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Wikipedia:Alain A. Lewis#0	Alain A. Lewis (born 1947) is an American mathematician. A student of the mathematical economist Kenneth Arrow, Lewis is credited by the historian of economics Philip Mirowski with making Arrow aware of computational limits to economic agency. == Life == Lewis gained his BA in philosophy, economics and statistics from George Washington University in 1969, and a PhD in applied mathematics from Harvard University in 1979. He was based at Lawrence Livermore Labs from 1978 to 1979, RAND from 1979 to 1982, the University of Singapore from 1981 to 1983, Cornell University from 1983 to 1987 and University of California, Irvine from 1987. == Works == 'A Nonstandard Theory of Games. Part I: On the Existence of the Quasi-Kernel and Related Solution Concepts for Finite Cooperative Games', Harvard University Center on Decision and Conflict in Complex Situations Technical Report no. TR-6, June 1979 'A Nonstandard Theory of Games. Part II. On Non-Atomic Representations', Harvard University, Technical Report no. TR-7, June 1979 'A Nonstandard Theory of Games. Part III. Noncooperative Finite Games', Harvard University, Technical Report no TR-8, June 1979 'A Nonstandard Theory of Games. Part IV. Equilibrium Points for Finite Games', Technical Report no. TR-9, June 1979 'Arrow's theorem and group decision making on public policy', RAND Papers, 1979 'Aspects of fair division', RAND Papers, 1980 (with Perry Thorndyke and others) 'Improving Training and Performance of Navy Teams: A Design for a Research Program', RAND Reports, 1980 'A note on the Lagrangean expression of Nash equilibria', RAND Papers, 1980 'On the formal character of plausible reasoning', RAND Papers, 1980 'The Use of Utility in Multiattribute Utility Analysis', RAND Papers, 1980 'Notes on finite cooperative games', RAND Papers, 1981 'Hyperfinite Von Neumann games', Mathematical Social Sciences, Vol. 9, No. 2 (1985), pp. 189–194 'Loeb-measurable solutions to finite games', Vol. 9, No. 3 (1985), pp. 197–247 'On effectively computable realizations of choice functions', Mathematical Social Sciences, Vol. 10, No. 1 (1985), pp. 43–80 'The minimum degree of recursively representable choice functions', Mathematical Social Sciences, Vol. 10, No. 2 (1985), pp. 179–88 'Complex structures and composite models – An essay on methodology', Mathematical Social Sciences, Vol. 10, No. 3 (1985), pp. 211–246 'On the construction of subinvariant weakly additive set-functions', Mathematical Social Sciences, Vol. 13, No. 1 (1987), pp. 81–86 'Some aspects of constructive mathematics that are relevant to the foundations of neoclassical mathematical economics and the theory of games', Stanford University Center for Research on Organization Efficiency, Technical Report No. 526 (April 1988). (with Rangarajan Sundaram) 'An alternate approach to axiomatizations of the von Neumann/Morgenstern characteristic function, Mathematical Social Sciences, Vol. 15, No. 2 (1988), pp. 145–56 'Lower bounds on degrees of game-theoretic structures', Mathematical Social Sciences, Vol. 16, No. 1 (1988), pp. 1–39 'An infinite version of arrow's theorem in the effective setting', Mathematical Social Sciences, Vol. 16, No. 1 (1988), pp. 41–48 'On the independence of core-equivalence results from Zermelo–Fraenkel set theory', Mathematical Social Sciences, Vol. 19. No. 1 (1990), pp. 55–95 'A note on degrees of presentation of games as relational structures', Mathematical Social Sciences, Vol. 19. No. 2 (1990), pp. 195–201 'A game-theoretic equivalence to the Hahn–Banach theorem', Mathematical Social Sciences, Vol. 20, No. 3 (1990), pp. 199–214 'On the Effective Content of Asymptotic Verifications of Edgeworth's Conjecture', 1991 (with Y. Inagaki) 'On the Effective Content of Theories', preprint, University of California at Irvine, School of Social Sciences, 1991. 'On Turing degrees of Walrasian models and a general impossibility result in the theory of decision-making', Mathematical Social Sciences, Vol. 24, No. 2-3 (1992), pp. 141–171 'Some aspects of effectively constructive mathematics that are relevant to the foundations of neoclassical mathematical economics and theory of games', Mathematical Social Sciences, Vol. 24, No. 2-3 (1992), pp. 209–235 == External links == "Most widely held works by Alain A Lewis". WorldCat. == References ==
Wikipedia:Alain Goriely#0	Alain Goriely is a Belgian mathematician, currently holding the statutory professorship (chair) of mathematical modelling at the University of Oxford, Mathematical Institute. He is director of the Oxford Centre for Industrial Mathematics (OCIAM), of the International Brain and Mechanics Lab (IBMTL) and Professorial Fellow at St Catherine's College, Oxford. At the Mathematical Institute, he was the director of external relations and public engagement, from 2013 until 2022, initiating the Oxford Mathematics series of public lectures. In 2022, he was elected to the Royal Society, and Gresham Professor of Geometry at the Gresham College (London) in 2024. == Education and early life == Born and raised in Brussels, Goriely obtained his B.Sc. in 1989 and Ph.D. in 1994 from the Université libre de Bruxelles where he became lecturer in the Mathematics Department. Shortly after, he moved to the University of Arizona to take the positions of Research Associate (1994-1997), Assistant Professor (1998-2002), Associate Professor (2002-2007) and Professor (2007-2010). In Tucson, he also served as acting head for the Program in Applied Mathematics in 2006-2007 and 2007–2008. In 2010, he moved to Oxford to take up the inaugural chair of Mathematical Modelling and to become Director of the Oxford Centre for Collaborative Applied Mathematics (OCCAM). He is a Senior Fellow of the Oxford Martin School and received a M.A. in 2010 from the University of Oxford (by resolution). He has held a number of positions, including visiting professorships at the École Polytechnique Fédérale de Lausanne, the École normale supérieure (Paris), and the Pierre and Marie Curie University. He also held the Timoshenko professorial fellowship and the Poincaré visiting professorship at Stanford University, the Springer professorship at the University of California, Berkeley and the Distinguished Rothschild Visiting Fellowship at the Isaac Newton Institute. == Research and career == Goriely works in the field of applied mathematics and he is interested in a broad range of problems including dynamical systems; the mechanics of biological growth; the modelling of the brain, the theoretical foundations of mechanics; the dynamics of curves, knots, and rods; the modelling of cancer; the development of new photovoltaic devices; the modelling of lithium-ion batteries and, more generally the study and development of mathematical methods for applied sciences. === Differential equations and dynamical systems === In his doctoral research on singularities, integrability theory, and dynamical systems, he established deep connections between the analytic and geometric approaches of differential equations by showing that the local behavior of the solutions of differential equations in complex time is connected to their global geometric properties in phase space. In particular, he developed new tests to prove the integrability and non-integrability for systems of differential equations and discrete mappings, based on the so-called Painlevé expansions in complex time. More importantly, he derived a new form of the Melnikov distance from the local Painleve property that can be used to prove the existence of transverse homoclinic connections, thereby directly relating local multivaluedness in complex time to chaotic dynamics in real-time. He also gave sufficient conditions for the existence of open sets of initial conditions leading to finite-time singularities which cosmologists use to explore possible singularities in cosmological models (such as the expanding general-relativistic Friedmann universe, brane singularity). These results are summarized in his monograph. === Curves and filaments === Over the years, Goriely has made important contributions to the modeling and analysis of filaments. Elastic curves can be modeled through the Kirchhoff equations that take into account bending, shearing, and extension. Within this context, in 1998 he identified a new type of instability driven by curvature. He showed that a torsional instability of filaments under tension can result in the formation of structures with opposite chirality for which he coined the word tendril perversion. Other contributions in this area include a complete classification of static solutions, the discovery of new exact dynamical solutions for the Kirchhoff elastic rods, and the development of new geometric methods to prove stability through the positive definiteness of the second variation. With colleagues, he provided a complete classification of uniform equilibria, and built the first three-dimensional theory for the nonlinear dynamics of elastic tubes conveying a fluid, studied the twining of vines, proved the existence of compact waves traveling on nonlinear rods, the inversion of curvature in bacteria, the growth of stems, the mechanics of seed expulsion, the shape and mechanics of proteins, and a full theory of growing and remodeling elastic rods suitable to describe many biological structures. With colleagues, he used this framework to develop a theory of plant tropism that include multiple stimuli. === Morphoelasticity === Goriely has worked in the applications of nonlinear mechanics to the field of biological materials and biological growth. Through his work, he was central in the development of a general mechanical theory of biological growth. This theory, for which he coined the word morphoelasticity, deals with the physical forces and shapes generated during development, homeostasis, or pathology. At the mathematical level, it is based on the general theory of nonlinear anelasticity. While the basic theoretical framework was understood as early as 1994, in 2005 with Martine Ben Amar, he developed a general stability method for morphoelastic solids and demonstrated that patterns and instabilities can be driven exclusively through growth. He further expanded this aspect of his research to demonstrate the occurrence of growth-induced patterns in many biological and physiological systems such as fungi, bacteria, and microbial cellular blebbing. Together with Derek Moulton and Régis Chirat, he developed a theory to describe morphological patterns for seashells, such as spikes and commarginal ornamentation. His theory of morphoelasticity is developed in his 2017 monograph on growth. === Mathematical foundations of mechanics === Goriely made several contributions to the foundations of classical mechanics and nonlinear elasticity. With his collaborators, he has given a general exact theory of Euler buckling within three-dimensional nonlinear elasticity, developed new fundamental adscititious inequalities for materials exhibiting the negative Poynting effect, and studied the nonlinear dynamics of shear waves in elastic solids. Since 2012, he initiated, with Arash Yavari, a research programme related to the geometric foundations of mechanics for nonlinear solids. In the absence of defects, solids can be described through the mapping of a reference configuration in the Euclidean space to a current configuration that also sits in Euclidean space. In the presence of defects, the correct underlying mathematical structure that describes the reference configuration is a non-Euclidean manifold. These ideas, first presented in the work of Kazuo Kondo in the 1940s, were known by the mechanics community but had never been used directly to build an effective theory of continuous defects. In this fully geometric theory, first described in their 2012 paper, they show that pure dislocations, disclinations, and point defects are, respectively, associated with Weitzenbock, Riemann, and Weyl manifolds. Further, they used Cartan's moving frames theory to formulate a complete theory of defects which can be used to obtain exact solutions for a number of important problems in nonlinear dislocation theory and anelasticity. They used this theory to obtain the exact nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid. They also introduced the concept of discombinations to describe sources of incompatibility related to multiple origins (point, lines, and edge defects). === Energy and materials === Goriely has done work in the field of materials science and renewable energy, ionic liquids, nano-particles fabrication, supercapacitors, and lithium-ion batteries. In 2013, he initiated a collaboration with Henry Snaith on the development of a new generation of perovskite solar cell. In their 2014 paper, they developed a mathematical model to predict coverage and morphology during the annealing of a thin solid film of a perovskite absorber. This model predicts the optimum film thickness and annealing temperature ensuring that it has exactly the right degree of transparency. === Brain modeling === Since 2012, Goriely has done some work related to the brain modeling. With his collaborators, he has developed models for axon growth based on the combined mechanics of microtubules extension, growth cone connection,. At the tissue level, with his collaborators, he developed new constitutive models for brain tissue validated on multi-axial shear experiments using human brain tissues. This work forms the basis for his models of swelling initiation and propagation showing that the Donnan effect is not sufficient and that swelling is also caused by an osmotic pressure increase driven by non-permeating solutes released by necrotic cells. At the organ level, he proposed the first mechanical models of craniectomy and craniosynostosis through systematic mathematical modeling, analysis and computational simulations in fully segmented brain geometry and explained the thickness asymmetry between gyri and sulci first noted more than 100 years ago by Brodmann. More recently, they developed a model for dementia propagation and showed that atrophy could be modeled through a multiplicative decomposition of the deformation gradient coupling mass removal to toxic proteins and studied the related cognitive decay. === Publications === Goriely is the author of three books Goriely A. Integrability and nonintegrability of dynamical systems. World Scientific; 2001. Goriely A. The mathematics and mechanics of biological growth. Springer; 2017 Goriely A. Applied Mathematics: A very short introduction. Oxford University Press; 2017 His most cited papers are: Eperon GE, Burlakov VM, Docampo P, Goriely A, Snaith HJ. Morphological control for high performance, solution-processed planar heterojunction perovskite solar cells. Advanced Functional Materials. 2014 Jan;24(1):151-7. According to Google Scholar, it has been cited 1922 times. Saidaminov MI, Abdelhady AL, Murali B, Alarousu E, Burlakov VM, Peng W, Dursun I, Wang L, He Y, Maculan G, Goriely A. High-quality bulk hybrid perovskite single crystals within minutes by inverse temperature crystallization. Nature Communications. 2015 Jul 6;6(1):1-6. According to Google Scholar, this article has been cited 1159 times Noel NK, Abate A, Stranks SD, Parrott ES, Burlakov VM, Goriely A, Snaith HJ. Enhanced photoluminescence and solar cell performance via Lewis base passivation of organic–inorganic lead halide perovskites. ACS Nano. 2014 Oct 28;8(10):9815-21. According to Google Scholar, this article has been cited 1131 times Stranks SD, Burlakov VM, Leijtens T, Ball JM, Goriely A, Snaith HJ. Recombination kinetics in organic-inorganic perovskites: excitons, free charge, and subgap states. Physical Review Applied. 2014 Sep 11;2(3):034007. According to Google Scholar, this article has been cited 950 times Ben Amar M, Goriely A. Growth and instability in elastic tissues. Journal of the Mechanics and Physics of Solids. 2005 Oct 1;53(10):2284-319. According to Google Scholar, this article has been cited 353 times Goriely A, Geers MG, Holzapfel GA, Jayamohan J, Jérusalem A, Sivaloganathan S, Squier W, van Dommelen JA, Waters S, Kuhl E. Mechanics of the brain: perspectives, challenges, and opportunities. Biomechanics and Modeling in Mechanobiology. 2015 Oct;14(5):931-65. According to Google Scholar, this article has been cited 271 times Cangelosi R, Goriely A. Component retention in principal component analysis with application to cDNA microarray data. Biology Direct. 2007 Dec;2(1):1-21. According to Google Scholar, this article has been cited 255 times == References ==
Wikipedia:Alain M. Robert#0	Alain M. Robert is Honorary Professor at University of Neuchâtel. Robert received his PhD from the University of Neuchâtel in 1967, where he studied under Roger Bader. His dissertation Quelques Questions d'Espaces Vectoriels Topologiques concerned topological vector spaces. == Selected publications == Robert, Alain M.: A course in p-adic analysis. Graduate Texts in Mathematics, 198. Springer-Verlag, New York, 2000. MathSciNet reviewer Daniel Barsky wrote: "Robert's book is aimed at an intermediate level between the very specialized monographs and the elementary texts. It has no equal in the marketplace, because it covers practically all of p-adic analysis of one variable (except the rationality of the zeta function of an algebraic variety over a finite field and the theory of p-adic differential equations) and contains numerous results that were accessible only in articles or even in preprints" here. This book was cited over 300 times at Google Scholar in 2012. Robert, Alain: Introduction to the representation theory of compact and locally compact groups. London Mathematical Society Lecture Note Series, 80. Cambridge University Press, Cambridge-New York, 1983. Robert, Alain: Elliptic curves. Notes from postgraduate lectures given in Lausanne 1971/72. Lecture Notes in Mathematics, Vol. 326. Springer-Verlag, Berlin-New York, 1973. Gertsch, Anne; Robert, Alain M.: Some congruences concerning the Bell numbers. Simon Stevin 3 (1996), no. 4, 467–475. Robert, Alain: Nonstandard analysis. Translated from the French by the author. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. This book was cited 80 times at Google Scholar in 2012. Robert, Alain M.: Linear Algebra: Examples and Applications. World Scientific Publishing Company, Singapore, 2005 == See also == Influence of non-standard analysis == References == == External links == Listing at University of Neuchâtel's Institut de mathématiques
Wikipedia:Alain-Sol Sznitman#0	Alain-Sol Sznitman (born 13 December 1955) is a French and Swiss mathematician who works as a professor of mathematics at ETH Zurich. His research concerns probability theory and mathematical physics. Within the field of percolation theory, Sznitman introduced the study of random interlacements. == Education and career == Sznitman did his undergraduate studies at the École Normale Supérieure, and earned a Doctorat d'Etat in 1983 from Pierre and Marie Curie University, under the supervision of Jacques Neveu. He worked at the Courant Institute of Mathematical Sciences at New York University beginning in 1983 and was promoted to full professor there in 1990. He moved to ETH Zurich in 1991, and from 1995 to 1999 was director of the Institute for Mathematical Research at ETH Zurich. He is a dual citizen of France and Switzerland. == Recognition == In 1991, Sznitman won the Rollo Davidson Prize, given annually to an early-career probabilist, and in 1999 he won the Line and Michel Loève International Prize in Probability. He became a fellow of the Institute of Mathematical Statistics in 1997, and of the American Mathematical Society in 2012. He was an invited speaker at the International Congress of Mathematicians in 1998. == References ==
Wikipedia:Alan B. Tayler#0	Alan Breach Tayler (1931–1995) was a British applied mathematician and pioneer of "industrial mathematics". He was a Founding Fellow of St Catherine's College, Oxford (1959-1995), the initiator of the Oxford Study Groups with Industry in 1968 (which developed into the European Study Groups with Industry), a driving force behind the foundation of the European Consortium for Mathematics in Industry (ECMI) in 1985 and President of ECMI (1989), and the first Director of the Oxford Centre for Industrial and Applied Mathematics (OCIAM) (1989–1994). == Education == Tayler was privately educated at King's College School in Wimbledon, London and the University of Oxford where he was a student at Brasenose College, Oxford in 1951. After he was awarded a first class undergraduate degree in Mathematics he worked for a brief period in industry, before returning to academia to complete his Doctor of Philosophy (DPhil) degree supervised by George Temple in 1959. == Career and research == Tayler was a distinguished applied mathematician who made important contributions in a wide range of areas (notably lubrication theory, surface gravity waves and viscous dissipation), but his key contribution to science was as the driving force behind the establishment of what is often called "mathematics-in-industry" or "industrial mathematics" (i.e. the application of mathematical approaches to the modeling and analysis of a wide range of real-world problems) as a recognized scientific discipline in its own right. His philosophy is perfectly exemplified by the Oxford (now European) Study Groups with Industry which he and Professor Leslie Fox created in 1968 and are still going strong today. His approach to mathematical modelling is described in his monograph "Mathematical Models in Applied Mechanics", and is commemorated by the annual Alan Tayler Lecture held at the St Catherine's College, Oxford in November each year. During his career he supervised research of several notable applied mathematicians, including John Ockendon and John King. In 1959 Alan Tayler became a University Lecturer and Tutorial Fellow at St Catherine's Society, Oxford, and was involved in its transformation into St Catherine's College, Oxford in 1962, where remained for the rest of his career. He was devoted to the College, and held several of its major offices, and also to the Oxford University Rugby Football Club (OURFC), of which he was President from 1990 to 1995. === Public service === Tayler served as a school governor of the private Abingdon School from 1964 to 1983 and as vice-chairman of its governing board from 1972 to 1981. === Awards and honours === In 1982 Alan Tayler was jointly awarded (with James Lighthill) the IMA Gold Medal of the Institute of Mathematics and its Applications (IMA) for his services to applied mathematics, and in 1993 he was appointed Commander of the Order of the British Empire (CBE) in 1993 for his services to applied science and industry. == Personal life == Tayler was born in Mitcham, Surrey on 5 September 1931 and died in Ducklington, Oxfordshire on 29 January 1995. In 1955 he married June Earp and they had four daughters, one of whom pre-deceased him. == References ==
Wikipedia:Alan Cobham (mathematician)#0	Alan Belmont Cobham (4 November 1927 – 28 June 2011) was an American mathematician and computer scientist known for (with Jack Edmonds and Michael O. Rabin) inventing the notion of polynomial time and the complexity class P,[B] for Cobham's thesis stating that the problems that have practically usable computer solutions are characterized by having polynomial time,[B] and for Cobham's theorem on the sets of numbers that can be recognized by finite automata.[C] He also did foundational work on automatic sequences,[D] invented priority queues and studied them from the point of view of queueing theory,[A] and wrote a program for playing contract bridge that was at the time (in the mid-1980s) one of the best in the world. Cobham was a student at Oberlin College, the University of Chicago, the University of California, Berkeley, and the Massachusetts Institute of Technology, but did not complete a doctorate. He became an operations researcher for the United States Navy, a researcher for IBM Research at the Thomas J. Watson Research Center, and a professor and founding department chair of the computer science department at Wesleyan University. == Selected publications == == References ==
Wikipedia:Alan Gaius Ramsay McIntosh#0	Alan Gaius Ramsay McIntosh (* 1942 in Sydney, † August 8, 2016) was an Australian mathematician who dealt with analysis (harmonic analysis, partial differential equations). He was a professor at the Australian National University in Canberra. McIntosh studied at the University of New England with a bachelor's degree in 1962 (as a student he also received the University Medal ) and PhD in 1966 with Frantisek Wolf at the University of California, Berkeley, ( Representation of Accretive Bilinear Forms in Hilbert Space by Maximal Accretive Operator ). In Berkeley, he was also a student of Tosio Kato. As a post-doctoral student, he was at the Institute for Advanced Study and from 1967 he taught at Macquarie University and from 1999 at the Australian National University. In 2014 he became emeritus. McIntosh was involved in solving the Calderon conjecture in the theory of singular integral operators. In 2002, he solved with Pascal Auscher, Michael T. Lacey, Philipp Tchamitchian and Steve Hofmann the open Kato root problem for elliptic differential operators. He also deals with singular integral operators, boundary value problems of partial differential equations with applications (such as scattering theory of the Maxwell equations in irregular areas), spectral theory and functional calculus of operators in Banach spaces, analysis with Clifford algebras, barriers for the heat kernel equation and functional calculus for elliptic partial differential operators. In 1986 he became a fellow of the Australian Academy of Science, whose Hannan Medal he received in 2015. In 2002 he received the Moyal Medal from Macquarie University. == References == Mathematical Legacy Obituary notice, 2016, from the Australian Mathematical Society
Wikipedia:Albert Badrikian#0	Albert Badrikian (born January 11, 1933 in Lyon; died July 31, 1994 in a crevasse of the Bossons Glacier in the Mont Blanc massif) was a French mathematician and professor of mathematics at the Blaise Pascal University. Badrikian co-founded the renowned and influential summer school École d’Été de Probabilités de Saint-Flour with Paul-Louis Hennequin. He specialized in stochastic processes, measure theory, and ε-entropy in information theory. == Early Life and Education == Badrikian was born on January 11 in 1933 in Lyon, as the third child of a family that had immigrated from Armenia. He attended university in Lyon, where he wrote two thesis papers, including one on stochastic processes in Banach spaces, and completed his Diplôme d’études supérieures spécialisées in 1952. Afterward, he worked for several years as an assistant and teacher in secondary schools. Afterwards he became a research intern at the CNRS in the Laboratoire de Probabilités at Université Paris IV in Paris, under the guidance of Robert Fortet. He served in the French military from 1960 to 1962. After returning to the CNRS in 1962, he began his doctoral studies under Fortet's supervision. During this period, he also sought advice from Laurent Schwartz. On December 18, 1967, he defended his dissertation Les éléments aléatoires vectoriels et leurs fonctionnelles caractéristiques and received the CNRS Bronze Medal for his work. In the late 1965 Badrikian received an invitation to the Université Blaise Pascal in Clermont-Ferrand, where he became a lecturer in 1968, a professor without chair in 1970, and a titular professor in 1972. He was instrumental in acquiring many books from the Russian school of mathematics for the university library. It was also particularly important to him to introduce young students to research. Between 1967 and 1969, he published articles on cylindrical measures and linear random functions, which were later compiled into a book titled Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques in 1970. During this time, French probabilists had frequent interactions, and probability theorists from Clermont-Ferrand, Lyon, and Dijon met quarterly. Badrikian also attended the NATO summer school on probabilistic methods in analysis and was a guest lecturer at Université du Québec, University of Toronto, and in Vienna. From 1969 to 1970 Badrikian participated in the Laurent Schwartz seminar and gave five lectures there. In April 1971 he attended the Colloquium on Functional Analysis in Bordeaux, and in June 1973, he visited the Colloquium on Gaussian Processes in Strasbourg with Simon Chevet. In 1971, he co-founded the École d’Été de Probabilités de Saint-Flour with Paul-Louis Hennequin. In 1974, Badrikian held a course titled Prolegomenon to Probability Calculus in Banach Spaces, which was later published. In 1982 he lectured at University of Mossul in Iraq, and in the same year he was invited to speak at the international colloquium on Measure Theory and Its Applications at Université de Sherbrooke in Canada. He returned to Sherbrooke in 1986 and gave a lecture on stochastic analysis. In 1987, he visited Sherbrooke again, as well as the universities in Montreal and Ottawa. In 1988, he was invited to the Chinese-French Center for Mathematics in Wuhan, China, where he visited seven times before his death. In 1989, he gave a course on stochastic integration and stochastic differential equations at Universidad de Chile in Santiago de Chile. In 1990, he was promoted from Associate Professor to Extraordinary Professor. In 1991, he was invited by the Latin American School for Probability and Statistics to give a course. In 1992, he went to Wrocław in Poland to establish a scientific partnership. Badrikian passed away on July 31 in 1994 in a fall into a crevasse on the Glacier des Bossons while preparing for a new ascent of the Mont Blanc massif. Badrikian was the doctoral advisor of Pierre Bernard. == Awards == CNRS Bronze Medal Member of the American Mathematical Society. == Publications (Selection) == Propriétés permanentes des trajectoires de processus, in Colloque d'analyse fonctionnelle (Bordeaux, 1971), Mémoires de la Société Mathématique de France, no. 31-32 (1972), pp. 21-29. doi:10.24033/msmf.59 with S. Chevet: Questions liées à la théorie des espaces de Wiener. Colloque sur les processus Gaussiens et sur les distributions aléatoires. Strasbourg 1973. Annales de l'Institut Fourier, volume 24 (1974) no. 2, pp. 1-25. Prolégomènes au calcul des probabilités dans les Banach. Ecole d’Eté de Saint-Flour (1975). Springer-Berlin (L.N.539) (1976) Fonctions convexes et mesures cylindriques. Conv. sull misure su gruppi e su spazi vect. Rome 1975. Academic Press, London (1977) pp.139-176. Transformation of Gaussian Measures. Ecole Latino Americaine de Proba. et Stats., Santiago du Chili 1991. (dans cet ouvrage pp.13-58). Calcul stochastique anticipatif par rapport à une mesure Gaussienne. Séminaire d’analyse moderne, Sherbrooke, Juillet 1986. Université de Sherbrooke (1988) 270p with G. W. Johnson and Il Yoo: The composition of operator-valued measurable functions is measurable. Proc. A.M.S., Vol.123, n°6, Juin 1995, pp.1815-1820. with Ali S. Üstünel: Radonification of cylindrical semimartingales on Hilbert spaces. Annales mathématiques Blaise Pascal, volume 3 (1996) no. 1, pp. 13-21. Measurable linear mappings from a Wiener space. Annales mathématiques Blaise Pascal, volume S3 (1996), pp. 59-113. Martingales hilbertiennes. Annales mathématiques Blaise Pascal, volume S3 (1996), pp. 115-171. === Books === Séminaire sur les Fonctions Aléatoires Linéaires et les Mesures Cylindriques (1970). Lecture Notes in Mathematics, vol 139. Springer, Berlin, Heidelberg. doi:10.1007/BFb0067894 Mesures Cylindriques, Espaces de Wiener et Fonctions Aléatoires Gaussiennes (1974). Lecture Notes in Mathematics, vol 379. Springer, Berlin, Heidelberg. doi:10.1007/BFb0060494 with Jia-An Yan and Paul-André Meyer: Probability And Statistics: French-Chinese Meeting - Proceedings Of The Wuhan Meeting. (1993). Singapore: World Scientific Publishing Company. == Bibliography == Hennequin, Paul-Louis (1996). "L'œuvre scientifique d'Albert Badrikian". Annales mathématiques Blaise Pascal (in French). S3: 1–11. == References ==
Wikipedia:Albert Châtelet#0	Albert Châtelet (24 October 1883 – 30 June 1960) was a French politician and mathematician. == Biography == Châtelet was a student at the École normale supérieure (Paris) from 1905 to 1908, succeeding to the Agrégation (a highly selective competitive examination for future high-school teachers) in 1908. After earning a doctorate in 1911 and serving first in the health service, then in a ballistic research unit during the First World War, Châtelet became a lecturer at École centrale de Lille and in 1920 a professor at Université de Lille, rising to the rank of vice-chancellor by 1924. After thirteen years of chancellorship he was appointed as the director of secondary education by the Ministry of National Education, where he served under Jean Zay until 1940. In 1945 he joined the Faculty of Science at the University of Paris, succeeding Jean Cabannes as its dean in 1949. After his retirement as dean in 1954, Châtelet began participating in political movements at the forefront of the downfall of the French Fourth Republic by joining the Rationalist Union in 1955. In 1958 Albert Châtelet was chosen to represent the Union of Democratic Forces as its candidate during the French presidential election. He earned only 8.4% of the vote, losing out to the Union of Democrats for the Republic candidate Charles de Gaulle. In 1947 Châtelet introduced a concept of normality for relations in composition series and proved a general theorem similar to Jordan–Hölder decomposition and Schreier refinement theorem. He also wrote "Algebra de relations de congruence". Investigation of binary relations was further pursued by his student Jacques Riguet. Châtelet also did research on number theory and group theory. He introduced the research of Kurt Hensel, Helmut Hasse, and the German school of p-adic number theorists into France. Châtelet edited volume 5 of the collected works of Henri Poincaré. In 1920 he gave a plenary address at the International Congress of Mathematicians in Strasbourg: Loi de Réciprocité Abélienne. In 1947 he was the president of the Société Mathématique de France. A university center in the 5th Arrondissement of Paris and a prize of the CNRS (Albert-Châtelet-Medaille) are named in his honor. One of his sons was the mathematician François Châtelet, who is not to be confused with the French philosopher of the same name. == References == Condette, Jean-François (2009), Albert Châtelet, la République par l'école (1883–1960) (in French), Artois Presses Université, ISBN 978-2-84832-106-6 Albert Châtelet Albert Châtelet at the Mathematics Genealogy Project
Wikipedia:Albert G. Howson#0	Albert Geoffrey Howson (1931 – 1 November 2022) was a British mathematician and educationist. He started to work as algebraist and in 1954 published the Howson property of groups and proved it for some types of groups. Later he devoted himself to the mathematics education and participated in reforms of mathematics education in the Great Britain and internationally. He was the editor-in-chief and chairman of Trustees of the School Mathematics Project in Great Britain and was involved in many other national and international projects. He worked at University of Southampton as head of the Department of Mathematics and Dean of the Faculty of Mathematical Studies and served as president of the Mathematical Association of Great Britain, and two terms as Secretary of the International Commission on Mathematical Instruction. Howson died on 1 November 2022, aged 91. == References ==
Wikipedia:Albert Muchnik#0	Albert Abramovich Muchnik (2 January 1934 – 14 February 2019) was a Russian mathematician who worked in the field of foundations and mathematical logic. == Biography == He received his Ph.D. from Moscow State Pedagogical Institute in 1959 under the advisorship of Pyotr Novikov. From there, he wrote his dissertation titled Solution to the Post Reducibility Problem. Muchnik's most significant contribution was on the subject of relative computability. He and Richard Friedberg independently introduced the priority method which gave an affirmative answer to Post's problem regarding the existence of recursively enumerable Turing degrees between 0 and 0' . This result, now known as the Friedberg–Muchnik theorem, opened study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure. Muchnik also made significant contributions to Medvedev's theory of mass problems, introducing a generalisation of Turing degrees, called "Muchnik degrees", in 1963. Muchnik also elaborated Kolmogorov's proposal of viewing intuitionism as "calculus of problems" and proved that the lattice of Muchnik degrees is Brouwerian. Muchnik was married to the Russian mathematician Nadezhda Ermolaeva. Their son Andrey Muchnik, who died in 2007, was also a mathematician working in foundations of mathematics. Albert Muchnik died in February 2019. == Selected publications == A. A. Muchnik, On the unsolvability of the problem of reducibility in the theory of algorithms. (in Russian) Doklady Akademii Nauk SSSR (N.S.), vol. 108 (1956), pp. 194–197 == References == == External links == Albert Mucknik's personal webpage,(in Russian) Keldysh Institute of Applied Mathematics
Wikipedia:Albert Nijenhuis#0	Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics. His high school studies at the gymnasium in Arnhem were interrupted by the evacuation of Arnhem by the Nazis after the failure of Operation Market Garden by the Allies. He continued his high school mathematical studies by himself on his grandparents’ farm, and then took state exams in 1945. His university studies were carried out at the University of Amsterdam, where he received the degree of Candidaat (equivalent to a Bachelor of Science) in 1947, and a Doctorandus (equivalent to a Masters in Science) in 1950, cum laude. He was a Medewerker (associate) at the Mathematisch Centrum (now the Centrum Wiskunde & Informatica) in Amsterdam 1951–1952. He obtained a PhD in mathematics in 1952, cum laude (Theory of the geometric object). His thesis advisor was Jan Arnoldus Schouten. He came to the United States in 1952 as a Fulbright fellow (1952–1953) at Princeton University. He then studied at the Institute for Advanced Study in Princeton, New Jersey 1953–1955, after which he spent a year as an Instructor in mathematics at the University of Chicago. He then moved to the University of Washington in Seattle, first as an assistant professor and then a professor of mathematics, departing in 1963 for the University of Pennsylvania, where he was a professor of mathematics until his retirement in 1987. He was a Fulbright Professor at the University of Amsterdam in 1963–1964, and a visiting professor at the University of Geneva in 1967–1968, and at Dartmouth College in 1977–1978. Following his retirement, he was a professor emeritus of the University of Pennsylvania and an Affiliate Professor at the University of Washington. In 1958 he was an invited speaker at the International Mathematical Congress in Edinburgh. He was a J.S. Guggenheim Fellow in 1961–1962, again studying at the Institute for Advanced Study. In 1966 he became a correspondent member of the Royal Netherlands Academy of Arts and Sciences, and in 2012 he became a fellow of the American Mathematical Society. == Career == His early work was in the area of differential geometry. He developed the Nijenhuis tensor in 1951, during his PhD studies at the University of Amsterdam. It was also during this time that he explored the properties of the Schouten-Nijenhuis bracket, although his work was not published until 1955. In a lecture at the American Mathematical Society Summer Institute in Differential Geometry (1956) in Seattle he was the first to mention deformations of complex structures and their exact relationship to cohomology. With Alfred Frölicher, he developed the Frölicher-Nijenhuis bracket (1955). Further work in this area with Roger Richardson yielded the Nijenhuis–Richardson bracket (1964). Soon thereafter his interests shifted to combinatorics. Much of his work was done with Herbert S. Wilf, with whom he published a book in 1975. After retiring, his interest in differential geometry was rekindled. His last conference presentation and paper were presented when he was nearly 70. == Personal life == Albert Nijenhuis became a U.S. citizen in 1959. He was married since 1955 and had four children. He died at the age of 88 after several months of failing health. == Selected publications == Nijenhuis, Albert; Richardson Jr., Roger W. (1966). "Cohomology and deformations in graded Lie algebras". Bulletin of the American Mathematical Society. 72 (1): 1–29. doi:10.1090/s0002-9904-1966-11401-5. MR 0195995. Nijenhuis, Albert (1972). Natural bundles and their general properties. Tokyo: Diff. Geom. in Honour of K. Yano. pp. 317–334. Nijenhuis, Albert; Wilf, Herbert S (1975). Combinatorial Algorithms. Academic Press. ISBN 0-12-519250-9. Nijenhuis, Albert; Wilf, Herbert S (1978). Combinatorial Algorithms for Computers and Calculators. Academic Press. ISBN 0-12-519260-6. Nijenhuis, Albert (1995). "Connection-free differential geometry". Diff. Geom. Appl., Proc. Conf.: 171–190. == References == == External links == Albert Nijenhuis at the Mathematics Genealogy Project University of Washington page Archived 2012-02-06 at the Wayback Machine
Wikipedia:Albert Pfluger#0	Albert Pfluger (13 October 1907, Oensingen – 14 September 1993, Zürich) was a Swiss mathematician, specializing in complex function theory. == Biography == Pfluger, the son of a farmer, attended secondary school in Stans. He then studied mathematics at ETH Zürich, where in 1935 he received his promotion (Ph.D) under George Pólya with thesis Über eine Interpretation gewisser Konvergenz- und Fortsetzungseigenschaften Dirichlet'scher Reihen. Afterwards, he became a Gymnasium teacher at the canton school in Zug and the canton school in Solothurn. He received his habilitation qualification in 1938 and in 1939 became a professor extraordinarius for applied mathematics and mathematical physics at the University of Fribourg. At ETH Zürich he became a professor extraordinarius in 1940 and in 1943 a professor ordinarius in the professorial chair vacated by George Pólya in 1940. Pfluger retired from ETH Zürich in 1978 as professor emeritus. He did research on the value distribution theory of Rolf Nevanlinna, potential theory, conformal maps and quasiconformal mapping, as well as Riemann surfaces. In 1957 Pfluger's monograph Theorie der Riemannschen Flächen was published in Springer's series Grundlehren der mathematischen Wissenschaften. With Joseph Hersch, he introduced in 1952 a function (now called the Hersch-Pfluger distortion function) useful in estimating the distortion of quasiconformal mappings. Entire functions with certain regularity properties were introduced and extensively studied by Pfluger and Boris Levin; these functions are now referred to as functions of completely regular growth in the sense of Levin and Pfluger. Pfluger served a two-year term as president of the Swiss Mathematical Society in 1950–1951. In 1973 he was elected a foreign member of the Finnish Academy of Sciences. His doctoral students include Peter Henrici and Heinz Rutishauser. Pfluger was married since 1938 to Maria Jeger. == Selected publications == with G. Pólya: "On the power series of an integral function having an exceptional value." In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31, no. 2, pp. 153–155. Cambridge University Press, 1935. doi:10.1017/S0305004100013244 "On analytic functions bounded at the lattice points." Proceedings of the London Mathematical Society 2, no. 1 (1937): 305–315. doi:10.1112/plms/s2-42.1.305 "Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen." Commentarii Mathematici Helvetici 11 (1938): 180–214. "Zur Defektrelation ganzer Funktionen endlicher Ordnung." Commentarii Mathematici Helvetici 19, no. 1 (1946): 91–104. doi:10.1007/BF02565950 "Quasikonforme Abbildungen und logarithmische Kapazität." Ann. Inst. Fourier 2 (1950): 69–80. doi:10.5802/aif.20 "Zur Stabilität des tangential gedrückten Stabes." ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 35, no. 5 (1955): 191. doi:10.1002/zamm.19550350506 "Extremallängen und Kapazität." Commentarii Mathematici Helvetici 29, no. 1 (1955): 120–131. doi:10.1007/BF02564275 "Über die Äquivalenz der geometrischen und der analytischen Definition quasikonformer Abbildungen." Commentarii Mathematici Helvetici 33, no. 1 (1959): 23–33. doi:10.1007/BF02565903 == Sources == Hans Künzi, Zum Gedenken an Albert Pfluger, 1907-1993. Elemente der Mathematik, Band 49, Heft 3, S. 89–91, 1994. Joseph Hersch (ed.): Complex analysis: articles dedicated to Albert Pfluger on the occasion of his 80. birthday. Birkhäuser 1988 == References ==
Wikipedia:Albert Shiryaev#0	Albert Nikolayevich Shiryaev (Russian: Альбе́рт Никола́евич Ширя́ев; born October 12, 1934) is a Soviet and Russian mathematician. He is known for his work in probability theory, statistics and financial mathematics. == Career == He graduated from Moscow State University in 1957. From that time until now he has been working in Steklov Mathematical Institute. He earned his candidate degree in 1961 (Andrey Kolmogorov was his advisor) and a doctoral degree in 1967 for his work "On statistical sequential analysis". He is a professor of the department of mechanics and mathematics of Moscow State University, since 1971. As of 2007 Shiryaev holds a 20% permanent professorial position at the School of Mathematics, University of Manchester. He has supervised more than 50 doctoral dissertations and is the author or coauthor of more than 250 publications. In 1970 he was an Invited Speaker with talk Sur les equations stochastiques aux dérivées partielles at the International Congress of Mathematicians (ICM) in Nice. In 1978 he was a Plenary Speaker with talk Absolute Continuity and Singularity of Probability Measures in Functional Spaces at the ICM in Helsinki. He was elected in 1985 an honorary member of the Royal Statistical Society and in 1990 a member of Academia Europaea. From 1989 to 1991 he was the president of the Bernoulli Society for Mathematical Statistics and Probability. From 1994 to 1998 he was the president of the Russian Actuarial Society. In 1996 he was awarded a Humboldt Prize. He was elected a corresponding member of the Russian Academy of Sciences in 1997 and a full member in 2011. From 1998 to 1999 he was a founding member and the first president of the Bachelier Finance Society. He was made in 2000 Doctor Rerum Naturalium Honoris Causa of Albert Ludwigs University of Freiburg and in 2002 Professor Honoris Causa of the University of Amsterdam. In 2017 he was awarded the Chebyschev gold medal of the Russian Academy of Sciences. == Contributions == His scientific work concerns different aspects of probability theory, statistics and its applications. He has contributions to: Nonlinear theory of stationary stochastic processes Problems of fast detection of random effects (Kolmogorov Prize of Russian Academy of Sciences, 1994) Problems of optimal nonlinear filtration, stochastic differential equations (A.N. Markov Prize of USSR Academy of Sciences, 1974) Problems of stochastic optimization, including "Optimal stopping rules" Problems of general stochastic theory and martingale theory Problems of stochastic finance (monograph "The Essentials of Stochastic Finance", English and Russian editions) == Publications == Optimal Stopping Rules. Springer-Verlag. 1978. ISBN 9783540740100. Statistical sequential analysis: optimal stopping rules. American Mathematical Society 1976 (Russian 1969), new edition entitled Optimal Stopping Rules, Springer 1978, 2008 with Robert Liptser: Statistics of random processes. 2 vols., Springer, 1977/1978, 1981; 2nd edition 2013, vol. 1 with P. Greenwood: Contiguity and Statistical Invariance Principle. Gordon and Breach, 1985 with R. Liptser: Theory of Martingales. Kluwer 1986; 2012 edition Probability (2nd ed.). Springer. 2013. ISBN 9781475725391; translated by R. P. Boas{{cite book}}: CS1 maint: postscript (link); 1st Russian edition 1980; 2nd Russian edition 1989, 2004 Wahrscheinlichkeit (= Hochschulbücher für Mathematik. vol. 91). Deutscher Verlag der Wissenschaften, Berlin 1988, ISBN 3-326-00195-9 (English: Probability, Springer 1984, 1996) with Jean Jacod: Limit Theorems for Stochastic Processes. Springer, 1994; 2nd edition 2013 Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific Publishing. 1999. ISBN 9789810236052. with V. Spokoiny: Statistical Experiments and Decision. World Scientific 2000 with A. V. Bulinsky: Theory of Stochastic Processes. A course of lectures. Moscow 2003 (in Russian) From "Disorder" to Nonlinear Filtering and Martingale Theory. In: Bolibruch, Osipov, Sinai (eds.): Mathematical Events of the Twentieth Century. Springer 2006, pp. 371–397 (translated by R. Cooke) doi:10.1007/3-540-29462-7_18 with Ole Barndorff-Nielsen: Change of Time and Change of Measure (2nd ed.). World Scientific Publishing. 2015. ISBN 9789814678605. 1st edition 2010 Problems in Probability. Springer. 2016. ISBN 9783764324193. == References == == External links == Albert Shiryaev at the Mathematics Genealogy Project A.N. Shiryaev at MSU (in Russian) A.N. Shiryaev at RAS. A.N. Shiryaev at Math-Net.Ru.
Wikipedia:Albert Wilansky#0	Albert "Tommy" Wilansky (13 September 1921, St. John's, Newfoundland – 3 July 2017, Bethlehem, Pennsylvania) was a Canadian-American mathematician, known for introducing Smith numbers. == Biography == Wilansky was educated as an undergraduate at Dalhousie University, where he received an M.A. in mathematics in 1944. From 1944 to 1947 he was a graduate student at Brown University. In 1947 he received his Ph.D. with advisor Clarence Raymond Adams and dissertation An application of Banach linear functionals to the theory of summability. From 1948 until his official retirement in 1992, Wilansky was a faculty member of the mathematics department of Lehigh University. He was the university’s Distinguished Professor of Mathematics for the final 14 years of his tenure. During his 44 years at Lehigh he was a Fulbright visiting professor several times, at universities in Reading (1972–1973), London (1973), Tel Aviv (1981), and Berne (1981). Outside of academia he was a consultant for the Frankford Arsenal for the year 1957–1958. Wilansky did research in analysis, specializing in summability theory, linear topological spaces, Banach algebras, and functional analysis. He was the author of several books and the author or co-author of more than 80 articles. He lectured at over 50 different universities. In 1969 he received the Mathematical Association of America's Lester R. Ford Award for his 1968 article Spectral Decomposition of Matrices for High School Students. (The 1969 award was also given individually to 5 other mathematicians.) Wilansky was married to his first wife from 1947 until her death in 1969. They had two daughters. He had three step-daughters from his second marriage. He was a professional musician for a brief time as a young man and continued playing piano and clarinet and writing songs, often with his wives and daughters. == Selected publications == === Articles === Wilansky, Albert (1949). "An application of Banach linear functionals to summability". Transactions of the American Mathematical Society. 67 (1): 59–68. doi:10.1090/S0002-9947-1949-0032025-7. ISSN 0002-9947. —— (1949). "A necessary and sufficient condition that a summability method be stronger than convergence". Bulletin of the American Mathematical Society. 55 (10): 914–916. doi:10.1090/S0002-9904-1949-09307-2. ISSN 0002-9904. ——; Zeller, Karl (1955). "Summation of bounded divergent sequences, topological methods". Transactions of the American Mathematical Society. 78 (2): 501–509. doi:10.1090/S0002-9947-1955-0067220-7. ISSN 0002-9947. ——; Zeller, Karl (1963). "A biorthogonal system which is not a Toeplitz basis". Bulletin of the American Mathematical Society. 69 (5): 725–726. doi:10.1090/S0002-9904-1963-11003-4. ISSN 0002-9904. —— (1976). "On a characterization of barrelled spaces". Proceedings of the American Mathematical Society. 57 (2): 375. doi:10.1090/S0002-9939-1976-0412761-2. ISSN 0002-9939. Kalton, Nigel; —— (1976). "Tauberian operators on Banach spaces". Proceedings of the American Mathematical Society. 57 (2): 251–255. doi:10.1090/S0002-9939-1976-0473896-1. ISSN 0002-9939. Saxon, Stephen A.; —— (1977). "The equivalence of some Banach space problems". Colloquium Mathematicum. 37 (2). Institute of Mathematics Polish Academy of Sciences: 217–226. doi:10.4064/cm-37-2-217-226. Snyder, A. K.; —— (1980). "The Mazur-Orlicz bounded consistency theorem". Proceedings of the American Mathematical Society. 80 (2): 374–376. doi:10.1090/S0002-9939-1980-0577777-5. ISSN 0002-9939. —— (1981). "Mazur spaces". International Journal of Mathematics and Mathematical Sciences. 4: 39–53. doi:10.1155/S0161171281000021. === Books === Functional analysis. New York: Blaisdell. 1964. Topics in functional analysis. Springer-Verlag. 1967; notes by W. D. Laverell{{cite book}}: CS1 maint: postscript (link) Wilansky, Albert (14 November 2006). 2006 pbk edition. Springer. ISBN 978-3-540-35525-0. Topology for analysis. Waltham, Massachusetts: Ginn. 1970. Wilansky, Albert (2008). Dover reprint. Courier Corporation. ISBN 9780486469034. Modern methods in topological vector spaces. New York: McGraw-Hill. 1978. Wilansky, Albert (2013). Dover reprint. Courier Corporation. ISBN 9780486493534. Summability through functional analysis. North-Holland. 1984. Wilansky, A. (April 2000). 2000 pbk edition. Elsevier. ISBN 9780080871967. == References ==
Wikipedia:Alberto Fujimori#0	Alberto Kenya Fujimori Inomoto (26 July 1938 – 11 September 2024) was a Peruvian politician, professor, and engineer who served as the 54th president of Peru from 1990 to 2000. Born in Lima, Fujimori was the country's first president of Japanese descent, and was an agronomist and university rector prior to entering politics. Fujimori emerged as a politician during the midst of the internal conflict in Peru, the Peruvian Lost Decade, and the ensuing violence caused by the far-left guerilla group Shining Path. In office as president, Fujimori implemented a series of military reforms and responded to Shining Path with repressive and lethal force, successfully halting the group's actions. His economic policy and his neoliberal political ideology of Fujimorism rescued Peru's economy and transformed its governance in the midst of its internal conflict. In 1992, during his first presidential term, Fujimori, with the support of the National Intelligence Service and the Peruvian Armed Forces, adopted Plan Verde and carried out a self-coup against the Peruvian legislature and judiciary. Fujimori dissolved the Peruvian Congress and Supreme Court, effectively making him a de facto dictator of Peru. The coup was criticized by Peruvian politicians, intellectuals and journalists, but was well received by the country's private business sector and a substantial part of the public. Following the coup d'état, Fujimori drafted a new constitution in 1993, which was approved in a referendum, and was elected as president for a second term in 1995 and controversially for a third term in 2000. Fujimori's time in office was marked by severe authoritarian measures, excessive use of propaganda, entrenched political corruption, multiple cases of extrajudicial killings, and human rights violations. Under the provisions of Plan Verde, Fujimori targeted members of Peru's indigenous community and subjected them to forced sterilizations. In 2000, following his third term election, Fujimori was facing mounting allegations of widespread corruption and crimes against humanity, in his government. Subsequently Fujimori fled to Japan, where he submitted his presidential resignation via fax. Peru's congress refused to accept his resignation, instead voting to remove him from office on the grounds that he was "permanently morally disabled". While in Japan, Peru issued multiple criminal charges against him, stemming from the corruption and human rights abuses that occurred during his government. Peru requested Fujimori's extradition from Japan, which was refused by the Japanese government due to Fujimori being a Japanese citizen, and Japanese laws stipulating against extraditing its citizens. In 2005, while Fujimori was visiting Santiago, Chile, he was arrested by the Carabineros de Chile by the request of Peru, and extradited to Lima to face charges in 2007. Fujimori was sentenced to 25 years in prison, but was unlawfully pardoned by president Pedro Pablo Kuczynski in 2017, and was officially released in December 2023. == Early life, education, and career == Alberto Kenya Fujimori Inomoto was born on 26 July 1938 in the Miraflores district of Lima, Peru, to Japanese parents Naoichi Fujimori (né Minami) and Mutsue Inomoto. His parents were originally from Kumamoto Prefecture and immigrated to Peru in 1934. Fujimori's parents were Buddhists, but he was baptized and raised Catholic. Aside from Spanish, he also spoke Japanese, the primary language in his childhood home. Fujimori obtained his early education at the Colegio Nuestra Señora de la Merced and La Rectora School. In 1956, he graduated from La Gran Unidad Escolar Alfonso Ugarte in Lima. Fujimori pursued his undergraduate studies at the Universidad Nacional Agraria La Molina in 1957, graduating first in his class in 1961 with a degree in agricultural engineering. He briefly lectured in mathematics at the university before moving to France to study physics at the University of Strasbourg. In 1969, he earned a master's degree in mathematics from the University of Wisconsin–Milwaukee through a Ford Foundation scholarship. In recognition of his academic achievements, the sciences faculty of the National Agrarian University offered Fujimori the deanship and in 1984 appointed him to the rectorship of the university, which he held until 1989. In 1987, Fujimori also became president of the National Assembly of University Rectors, a position that he held twice. He also hosted a TV show called Concertando from 1988 to 1989 on Peru's state-owned network, Canal 7. === Birthplace disputation === In July 1997, the news magazine Caretas alleged that Fujimori was born in Japan, in his father's hometown of Kawachi, Kumamoto Prefecture. Because the Constitution of Peru requires the president to have been born in Peru, this would have made Fujimori ineligible to be president. The magazine, which had been sued for libel by Vladimiro Montesinos seven years earlier, reported that Fujimori's birth and baptismal certificates might have been altered. Caretas also alleged that Fujimori's mother declared having two children when she entered Peru; Fujimori was the second of four children. Caretas's contentions were hotly contested in the Peruvian media; the magazine Sí described the allegations as "pathetic" and "a dark page for [Peruvian] journalism". Latin American scholars Cynthia McClintock and Fabián Vallas note that the issue appeared to have died down among Peruvians after the Japanese government announced in 2000 that "Fujimori's parents had registered his birth in the Japanese Embassy in Lima". The Japanese government determined that he was also a Japanese citizen because of his parents' registration in the koseki. == Presidency (1990–2000) == === First term === ==== 1990 general election ==== During the first presidency of Alan García, the economy had entered a period of hyperinflation and the political system was in crisis due to the country's internal conflict, leaving Peru in "economic and political chaos". The armed forces grew frustrated with the inability of the García administration to handle the nation's crises and began to draft Plan Verde as a plan to overthrow his government. According to Rospigliosi, lawyer and friend of Fujimori, Vladimiro Montesinos was not initially involved with the Plan Verde, but his ability to resolve issues for the military resulted with the armed forces tasking Montesinos with implementing the plan with Fujimori, Both General Nicolás de Bari Hermoza and Montesinos were responsible for the relationship between the armed forces and Fujimori. Mario Vargas Llosa, Fujimori's final opponent in the election, later reported that United States Ambassador to Peru, Anthony C. E. Quainton, personally told him that allegedly leaked documents of the Central Intelligence Agency (CIA) purportedly being supportive of Fujimori's candidacy were authentic. Rendón writes that the United States supported Fujimori because of his relationship with Montesinos, who had previously been charged with spying on the Peruvian military for the CIA. During the second round of elections, Fujimori originally received support from left-wing groups and those close to the García government, exploiting the popular distrust of the existing Peruvian political establishment and the uncertainty about the proposed neoliberal economic reforms of his opponent, novelist Mario Vargas Llosa. Fujimori won the 1990 presidential election as a dark horse candidate under the banner of Cambio 90, defeating Vargas Llosa in a surprise result. He capitalized on profound disenchantment with outgoing president Alan García and the American Popular Revolutionary Alliance party (APRA). During the campaign, Fujimori was nicknamed "el chino," which translates to "the Chinese guy" or "the Chinaman"; it is common for people of any East Asian descent to be called chino in Peru, as elsewhere in Spanish-speaking Latin America, both derogatorily and affectionately. Although he was of Japanese heritage, Fujimori suggested that he was always pleased by the nickname, which he perceived as a term of affection. With his election victory, he became the third person of East Asian descent to serve as presidency of a South American state, after President Arthur Chung of Guyana and Henk Chin A Sen of Suriname. ==== Economic shock ==== According to news magazine Oiga, the armed forces finalized plans on 18 June 1990 involving multiple scenarios for a coup d'état to be executed on 27 July 1990, the day prior to Fujimori's inauguration. The magazine noted that in one of the scenarios, titled "Negotiation and agreement with Fujimori. Bases of negotiation: concept of directed Democracy and Market Economy", Fujimori was to be directed on accepting the military's plan at least 24 hours before his inauguration. Fernando Rospigliosi states "an understanding was established between Fujimori, Montesinos and some of the military officers" involved in the Plan Verde prior to Fujimori's inauguration. Montesinos and SIN officials ultimately assumed the armed forces' position in the plan, placing SIN operatives into military leadership roles. Fujimori went on to adopt many of the policies outlined in the Plan Verde. Fujimori was sworn in as president on 28 July 1990, allegedly his 52nd birthday. After taking office, Fujimori abandoned the economic platform he promoted during his campaign, adopting more aggressive neoliberal policies than those espoused by Vargas Llosa, his opponent in the election. During his first term in office, Fujimori enacted wide-ranging neoliberal reforms, known as the Fujishock. It was Fujimori's stated objective to pacify the nation and restore economic balance. This program bore little resemblance to his campaign platform and was in fact more drastic than anything Vargas Llosa had proposed. Hernando de Soto, the founder of one of the first neoliberal organizations in Latin America, the Institute for Liberty and Democracy (ILD), began to receive assistance from the U.S. government under Ronald Reagan, with the National Endowment for Democracy's Center for International Private Enterprise (CIPE) providing his ILD with funding and education for advertising campaigns. Between 1988 and 1995, de Soto and the ILD were mainly responsible for some four hundred initiatives, laws, and regulations that led to significant changes in Peru's economic system. Under Fujimori, de Soto served as "the President's personal representative", with The New York Times describing de Soto as an "overseas salesman" for Fujimori in 1990, writing that he had represented the government when meeting with creditors and United States representatives. Others dubbed de Soto as the "informal president" for Fujimori. De Soto proved to be influential to Fujimori, who began to repeat de Soto's advocacy for deregulating the Peruvian economy. The International Monetary Fund (IMF) was content with Peru's measures, and guaranteed loan funding for Peru. Inflation rapidly began to fall and foreign investment capital flooded in. Nonetheless, the Fujishock restored Peru to the global economy, though not without immediate social cost; international business participated in crony capitalism with the government. The privatization campaign involved selling off of hundreds of state-owned enterprises, and replacing the country's troubled currency, the inti, with the nuevo sol. Fujimori's initiative relaxed private sector price controls, drastically reduced government subsidies and government employment, eliminated all exchange controls, and also reduced restrictions on investment, imports, and capital. Tariffs were radically simplified, the minimum wage was immediately quadrupled, and the government established a US$400 million poverty relief fund. The latter seemed to anticipate the economic agony to come: the price of electricity quintupled, water prices rose eightfold, and gasoline prices 3,000%. ==== Military regime ==== During Fujimori's first term in office, APRA, Vargas Llosa's party, and the Democratic Front remained in control of both chambers of Congress—then composed of a Chamber of Deputies and a Senate—hampering the enactment of economic reform. Fujimori also had difficulty combating the Shining Path due largely to what he perceived as intransigence and obstructionism in Congress. By March 1992, the Congress met with the approval of only 17% of the electorate, according to one poll; in the same poll, the president's approval stood at 42%. Fujimori and his military handlers had planned for a coup during his preceding two years in office. In response to the political deadlock, Fujimori, with the support of the military, carried out a self-coup on 5 April 1992, Congress was shut down by the military, the constitution was suspended and the judiciary was dissolved. Without political obstacles, the military was able to implement the objectives outlined in Plan Verde while Fujimori served as president to project an image that Peru was supporting a liberal democracy. Vladimiro Montesinos would go on to adopt the actual function of Peru's government. The coup was well received by the public, with Fujimori's approval rating jumping significantly in the wake of the coup. Fujimori often cited this public support in defending the coup, which he characterized as "not a negation of real democracy, but on the contrary... a search for an authentic transformation to assure a legitimate and effective democracy". Fujimori believed that Peruvian democracy had been nothing more than "a deceptive formality—a façade". He claimed the coup was necessary to break with the deeply entrenched special interests that were hindering him from rescuing Peru from the chaotic state in which García had left it. Fujimori's coup was immediately met with near-unanimous condemnation from the international community. The Organization of American States (OAS) denounced the coup and demanded a return to "representative democracy", despite Fujimori's claim that the coup represented a "popular uprising". Foreign ministers of OAS member states reiterated this condemnation of the autogolpe. They proposed an urgent effort to promote the reestablishment of "the democratic institutional order" in Peru. Negotiations between the OAS, the government, and opposition groups initially led Fujimori to propose a referendum to ratify the auto-coup, but the OAS rejected this. Fujimori then proposed scheduling elections for a Democratic Constituent Congress (CCD), which would draft a new constitution to be ratified by a national referendum. Despite a lack of consensus among political forces in Peru regarding this proposal, an ad hoc OAS meeting of ministers nevertheless endorsed this scenario in mid-May. Elections for the Democratic Constituent Congress were held on 22 November 1992. Various states individually condemned the coup. Venezuela broke off diplomatic relations, and Argentina withdrew its ambassador. Chile joined Argentina in requesting Peru's suspension from the Organization of American States. International lenders delayed planned or projected loans, and the United States, Germany, and Spain suspended all non-humanitarian aid to Peru. Fujimori, in turn, later received most of the participants of the November 1992 Venezuelan coup attempt as political asylees, who had fled to Peru after its failure. Peru–United States relations earlier in Fujimori's presidency had been dominated by questions of coca eradication and Fujimori's initial reluctance to sign an accord to increase his military's eradication efforts in the lowlands. Fujimori's autogolpe became a major obstacle to relations, as the United States immediately suspended all military and economic aid, with exceptions for counter-narcotic and humanitarian funds. Two weeks after the self-coup, the George H. W. Bush administration changed its position and officially recognized Fujimori as the legitimate leader of Peru, partly because he was willing to implement economic austerity measures, but also because of his adamant opposition to the Shining Path. On 13 November 1992, General Jaime Salinas Sedó attempted to overthrow Fujimori in a failed military coup. Salinas asserted that his intentions were to turn Fujimori over to be tried for violating the constitution. === Second term === The 1993 Constitution allowed Fujimori to run for a second term, and in April 1995, at the height of his popularity, Fujimori easily won reelection with almost two-thirds of the vote. His main opponent, former UN Secretary-General Javier Pérez de Cuéllar, won only 21 percent of the vote. Fujimori's supporters won comfortable majority in the new unicameral Congress. One of the first acts of the new congress was to declare an amnesty for all members of the military and police accused or convicted of human rights abuses between 1980 and 1995. During his second term, Fujimori and Ecuadorian President Sixto Durán Ballén signed a peace agreement over a border dispute that had simmered for more than a century. The treaty allowed the two countries to obtain international funds for developing the border region. Fujimori also settled some issues with Chile, Peru's southern neighbor, which had been unresolved since the 1929 Treaty of Lima. The 1995 election was the turning point in Fujimori's career. Peruvians began to be more concerned about freedom of speech and the press. Before he was sworn in for a second term, he stripped two universities of their autonomy and reshuffled the national electoral board. This led his opponents to call him "Chinochet", a reference to his previous nickname and to Chilean dictator Augusto Pinochet. Modeling his rule after Pinochet, Fujimori reportedly enjoyed this nickname. According to a poll by the Peruvian Research and Marketing Company conducted in 1997, 40.6% of Lima residents considered President Fujimori an authoritarian. In addition to the fate of democracy under Fujimori, Peruvians were becoming increasingly interested in the myriad allegations of criminality that involved Fujimori and his chief of the National Intelligence Service (SIN), Vladimiro Montesinos. Using SIN, Fujimori gained control of the majority of the armed forces, with the Financial Times stating that "[i]n no other country in Latin America did a president have so much control over the armed forces". A 2002 report by Health Minister Fernando Carbone later suggested that Fujimori was involved in the forced sterilizations of up to 300,000 indigenous women between 1996 and 2000, as part of a population control program. A 2004 World Bank publication said that in this period Montesinos's abuse of the power Fujimori granted him "led to a steady and systematic undermining of the rule of law". === Third term, flight to Japan and resignation === By the arrival of the new millennium, Alberto Fujimori became increasingly authoritarian, strengthening collaboration with Vladimiro Montesinos and the National Intelligence Service. Shortly after Fujimori began his second term, his supporters in Congress passed a law of "authentic interpretation" which effectively allowed him to run for another term in 2000. A 1998 effort to repeal this law by referendum failed. In late 1999, Fujimori announced that he would run for a third term. The electoral authorities, which were politically sympathetic to Fujimori, accepted his argument that the two-term restriction did not apply to him, as it was enacted while he was already in office. Exit polls showed Fujimori fell short of the 50% required to avoid an electoral runoff, but the first official results showed him with 49.6% of the vote, just short of outright victory. Eventually, Fujimori was credited with 49.9%—20,000 votes short of avoiding a runoff. Despite reports of numerous irregularities, the international observers recognized an adjusted victory of Fujimori. As voting is mandatory in Peru, Fujimori's primary opponent, Alejandro Toledo, called for his supporters to spoil their ballots in the runoff by writing "No to fraud!" on them. The OAS electoral observation mission pulled out of the country, saying that the process would be neither free nor fair. In the runoff, Fujimori won with 51.1% of the total votes. While votes for Toledo declined from 37.0% of the total votes cast in the first round to 17.7% of the votes in the second round, invalid votes jumped from 8.1% of the total votes cast in the first round to 31.1% of total votes in the second round. The large percentage of invalid votes in the election suggests widespread dissatisfaction with the electoral process among voters. Although Fujimori won the runoff with only a bare majority (but 3/4 valid votes), rumors of irregularities led most of the international community to shun his third swearing-in on 28 July. For the next seven weeks, there were daily demonstrations in front of the presidential palace. As a conciliatory gesture, Fujimori appointed former opposition candidate Federico Salas as prime minister. Opposition parties in Congress refused to support this move, and Toledo campaigned vigorously to have the election annulled. At this point, a corruption scandal involving Vladimiro Montesinos broke out, and exploded into full force on the evening of 14 September 2000, when the cable television station Canal N broadcast footage of Montesinos apparently bribing opposition congressman Alberto Kouri to defect to Fujimori's Peru 2000 party. The video was originally presented at press conference by Fernando Olivera and Luis Iberico of the FIM (Independent Moralizing Front); many other similar videos were released in the following weeks. Fujimori's support virtually collapsed, and a few days later he announced in a nationwide address that he would shut down the SIN and call new elections, in which he would not be a candidate. On 10 November, Fujimori won approval from Congress to hold elections on 8 April 2001. On 19 November, government ministers presented their resignations en bloc. Fujimori's first vice president, Francisco Tudela, had broken with Fujimori and resigned a few days earlier. This left second vice president Ricardo Márquez Flores as next in line for the presidency. Congress refused to recognize him, as he was an ardent Fujimori loyalist; Márquez resigned two days later. Paniagua was next in line, and became interim president to oversee the April 2001 elections. == Post-presidency (2000–2024) == === Summary === In 2000, facing charges of corruption and human rights abuses, Fujimori fled Peru and took refuge in Japan. He maintained a self-imposed exile until his arrest while visiting Chile in November 2005. He was extradited to face criminal charges in Peru on 22 September 2007. In December 2007, Fujimori was convicted of ordering an illegal search and seizure and was sentenced to six years imprisonment. The Supreme Court upheld the decision on appeal. In April 2009, Fujimori was convicted of human rights violations and sentenced to 25 years imprisonment for his role in kidnappings and murders by the Grupo Colina death squad during his government's battle against the Túpac Amaru Revolutionary Movement in the 1990s. Specifically, he was found guilty of murder, bodily harm and two cases of kidnapping. The verdict marked the first time that an elected head of state was tried and convicted of human rights violations. In July 2009, Fujimori was sentenced to 7+1⁄2 years imprisonment for embezzlement after he admitted to giving US$15 million from the Peruvian treasury to Montesinos. Two months later, he pleaded guilty in a fourth trial to bribery and received an additional six-year term. Transparency International determined the money embezzled by the Fujimori government—about US$600 million or about US$861 million in 2021—to be the seventh-most for a head of government active within 1984–2004. Under Peruvian law, all the resultant sentences must run concurrently; thus, the maximum length of imprisonment remained 25 years. In December 2017, Fujimori was pardoned by President Pedro Pablo Kuczynski, shortly after Fujimori's son, Congressman Kenji Fujimori, helped President Kuczynski survive an impeachment vote. The pardon was overturned by the Supreme Court on 3 October 2018, and Fujimori was sent back to prison in January 2019. The Constitutional Court of Peru in a 4–3 ruling on 17 March 2022 reinstated the pardon. On 8 April 2022, the Inter-American Court of Human Rights overruled the Constitutional Court and ordered Peru not to release Fujimori. The Constitutional Court ordered on 5 December 2023 that he be immediately released. === Resignation, arrest, and trial === On 13 November, Fujimori left Peru for a visit to Brunei to attend the Asia-Pacific Economic Cooperation forum. On 16 November, Valentín Paniagua took over as president of Congress after the pro-Fujimori leadership lost a vote of confidence. On 17 November, Fujimori traveled from Brunei to Tokyo, where he submitted his presidential resignation via fax. Congress refused to accept his resignation, instead voting on 22 November 62–9 to remove Fujimori from office on the grounds that he was "permanently morally disabled". After Congress rejected Fujimori's faxed resignation, they relieved Fujimori of his duties as president and banned him from Peruvian politics for a decade. Alejandro Toledo, who assumed the presidency in 2001, spearheaded the criminal case against Fujimori. He arranged meetings with the Supreme Court, tax authorities, and other powers in Peru to "coordinate the joint efforts to bring the criminal Fujimori from Japan". His vehemence in this matter at times compromised Peruvian law: forcing the judiciary and legislative system to keep guilty sentences without hearing Fujimori's defense; not providing Fujimori with representation when Fujimori was tried in absentia; and expelling pro-Fujimori congressmen from the parliament without proof of the accusations against them. Those expulsions were later reversed by the judiciary. Congress authorized charges against Fujimori in August 2001. Fujimori was alleged to be a coauthor, along with Vladimiro Montesinos, of the death-squad killings at Barrios Altos in 1991 and La Cantuta in 1992, respectively. At the behest of Peruvian authorities, in March 2003 Interpol issued an arrest order for Fujimori on charges that included murder, kidnapping, and crimes against humanity. In September 2003, Fujimori and several of his ministers were denounced for crimes against humanity, for allegedly having overseen forced sterilizations during his regime. In November, Congress approved an investigation of Fujimori's involvement in the airdrop of Kalashnikov rifles into the Colombian jungle in 1999 and 2000 for guerrillas of the Revolutionary Armed Forces of Colombia (FARC). Fujimori maintained he had no knowledge of the arms-trading, and blamed Montesinos. By approving the charges, Congress lifted the immunity granted to Fujimori as a former president, so that he could be criminally charged and prosecuted. Congress also voted to support charges against Fujimori for the detention and disappearance of 67 students from the central Andean city of Huancayo and the disappearance of several residents from the northern coastal town of Chimbote during the 1990s. It also approved charges that Fujimori mismanaged millions of dollars from Japanese charities, suggesting that the millions of dollars in his bank account were far too much to have been accumulated legally. In 2004, the Special Prosecutor established to investigate Fujimori released a report alleging that the Fujimori administration had obtained US$2 billion though graft. Most of this money came from Vladimiro Montesinos's web of corruption. The Special Prosecutor's figure of two billion dollars is considerably higher than the one arrived at by Transparency International, an NGO that studies corruption. Transparency International listed Fujimori as having embezzled an estimated US$600 million or about $861 million in 2021, which would rank seventh in the list of money embezzled by heads of government active within 1984–2004. Fujimori dismissed the judicial proceedings underway against him as "politically motivated", citing Toledo's involvement. Fujimori established a new political party in Peru, Sí Cumple, working from Japan. He hoped to participate in the 2006 presidential elections, but in February 2004, the Constitutional Court dismissed this possibility, because the ex-president was specifically barred by Congress from holding any office for ten years. Fujimori saw the decision as unconstitutional, as did his supporters such as former congress members Luz Salgado, Martha Chávez and Fernán Altuve, who argued it was a "political" maneuver and that the only body with the authority to determine the matter was the National Elections Jury (JNE). Valentín Paniagua disagreed, suggesting that the Constitutional Court finding was binding and that "no further debate is possible". Fujimori's Sí Cumple (roughly translated, "He Keeps His Word") received more than 10% in many country-level polls, contending with APRA for the second-place slot, but did not participate in the 2006 elections after its participation in the Alliance for the Future (initially thought as Alliance Sí Cumple) had not been allowed. Fujimori remained in self-imposed exile in Japan, where he resided with his friend, the Catholic novelist Ayako Sono. Several senior Japanese politicians supported Fujimori, partly because of his decisive action in ending the 1996–97 Japanese embassy crisis. Peru had requested Fujimori's extradition from Japan, which was refused by the Japanese government due to Fujimori being a Japanese citizen, and Japanese laws stipulating against extraditing its citizens. By March 2005, it appeared that Peru had all but abandoned its efforts to extradite Fujimori from Japan. In September of that year, Fujimori obtained a new Peruvian passport in Tokyo and announced his intention to run in the upcoming 2006 national election. Fujimori arrived in Chile in November 2005, but hours after his arrival there he was arrested following a arrest warrant issued by a Chilean judge, Peru then requested his extradition. While under house arrest in Chile, Fujimori announced plans to run in Japan's Upper House elections in July 2007 for the far-right People's New Party. Fujimori was extradited from Chile to Peru in September 2007. On 7 April 2009, a three-judge panel convicted Fujimori on charges of human rights abuses, declaring that the "charges against him have been proven beyond all reasonable doubt". The panel found him guilty of ordering the Grupo Colina death squad to commit the November 1991 Barrios Altos massacre and the July 1992 La Cantuta massacre, which resulted in the deaths of 25 people, as well as for taking part in the kidnappings of opposition journalist Gustavo Gorriti and businessman Samuel Dyer Ampudia. As of 2009 Fujimori's conviction is the only instance of a democratically elected head of state being tried and convicted of human rights abuses in his own country. Later on 7 April, the court sentenced Fujimori to 25 years in prison. Likewise, the Court found him guilty of aggravated kidnapping, under the aggravating circumstance of cruel treatment, with respect to journalist Gustavo Gorriti and businessman Samuel Dyer Ampudia. The Special Criminal Chamber determined that the sentence was to expire on 10 February 2032. On 2 January 2010, the sentence to 25 years in prison for human rights violations was confirmed. He faced a third trial in July 2009 over allegations that he illegally gave US$15 million in state funds to Vladimiro Montesinos, former head of the National Intelligence Service, during the two months prior to his fall from power. Fujimori admitted paying the money to Montesinos but claimed that he had later paid back the money to the state. On 20 July, the court found him guilty of embezzlement and sentenced him to a further 7+1⁄2 years in prison. A fourth trial took place in September 2009 in Lima. Fujimori was accused of using Montesinos to bribe and tap the phones of journalists, businessmen and opposition politicians—evidence of which led to the collapse of his government in 2000. Fujimori admitted the charges but claimed that the charges were made to damage his daughter's presidential election campaign. The prosecution asked the court to sentence Fujimori to eight years imprisonment with a fine of US$1.6 million plus US$1 million in compensation to ten people whose phones were bugged. Fujimori pleaded guilty and was sentenced to six years' imprisonment on 30 September 2009. === Pardon requests and release === Press reports in late 2012 indicated that Fujimori was suffering from tongue cancer and other medical problems. His family asked President Ollanta Humala for a pardon. President Humala rejected a pardon in 2013, saying that Fujimori's condition was not serious enough to warrant it. In July 2016, with three days left in his term, President Humala said that there was insufficient time to evaluate a second request to pardon Fujimori, leaving the decision to his successor Pedro Pablo Kuczynski. On 24 December 2017, President Kuczynski pardoned him on health grounds. Kuczynski's office stated that the hospitalized 79-year-old Fujimori had a "progressive, degenerative and incurable disease". The pardon kicked off at least two days of protests and led at least three congressmen to resign from Kuczynski's party. A spokesman for Popular Force alleged there was a pact that, in exchange for the pardon, Popular Force members helped Kuczynski fight ongoing impeachment proceedings. On 20 February 2018, the National Criminal Chamber ruled that it did not apply the resolution that granted Fujimori the right of grace for humanitarian reasons. Therefore, the former president had to face the process for the Pativilca Case with a simple appearance. On 3 October 2018, the Peruvian Supreme Court reversed Fujimori's pardon and ordered his return to prison. He was rushed to a hospital and returned to prison on 23 January 2019. His pardon was formally annulled on 13 February 2019. The Constitutional Court, in a 4–3 ruling on 17 March 2022, reinstated the pardon, though it was not clear if or when he might be released. Those ruling in approval of Fujimori's release argued that a pardon, no matter how unconstitutional it may be, can be issued by the President of Peru and that previous rulings annulling the pardon were "subjective". Constitutional Court judges ruling in favor of releasing Fujimori ignored the Inter-American Court of Human Rights' opinion that criticized Kuczynski's reported pardon pact with Fujimori's son and pointed out that the disease cited in the pardon was possibly diagnosed by Fujimori's personal doctor, not an independent physician. On 8 April 2022, the Inter-American Court of Human Rights overruled the Constitutional Court and ordered Peru not to release Fujimori. On 5 December 2023, he was ordered to be released immediately following an order by the Constitutional Court. This followed a previous order by the court that mandated the decision in the hands of a lower court in Ica, which returned the case to the Constitutional Court citing lack of authority. The following day, he was released from Barbadillo Prison in Lima, after spending 16 years in prison, whereupon he was met by his children Keiko and Kenji as well as a crowd of supporters. === Forced sterilizations trial === In May 2023, the Supreme Court of Chile ordered Fujimori to testify regarding forced sterilizations that occurred between 1996 and 2000 during his government, with Chile attempting to decide if they would expand extradition charges against Fujimori to include the sterilizations, which would allow him to be prosecuted in Peru. On 19 May 2023, Fujimori participated in a video call from Barbadillo Prison with justice officials in Chile defending his actions regarding sterilizations. === 2026 general election === Two months before his death, on 14 July 2024, Keiko Fujimori announced her father's candidacy for the 2026 Peruvian general election, despite his legal impediments and difficulties related to old age and poor health. == Personal life, illness, and death == In 1974, he married Susana Higuchi, also Japanese Peruvian. They had four children, including a daughter, Keiko, and a son, Kenji, who followed him into politics and were both elected to Congress. In 1994, Fujimori separated from Higuchi and formally stripped her of the title First Lady in August 1994, appointing Keiko as first lady in her stead. Higuchi publicly denounced Fujimori as a "tyrant" and claimed that his administration was corrupt. They formally divorced in 1995. For some years before his death, Fujimori had gastrointestinal issues, heart problems and cancer. He was in prison for several years following his presidency and was released on humanitarian grounds in December 2023. He was diagnosed with tongue cancer in early 2024. He made his last public appearance at a hospital after undergoing a CT scan on 4 September 2024. On 11 September, several Fujimorist members of congress wearing black, along with a priest, arrived at the home of Fujimori's daughter Keiko in Lima's San Borja District, amid reports that his health was failing. Subsequently, his doctor Alejandro Aguinaga told the press that he was "fighting" for his life and requested that visits be restricted. Congressperson Luisa María Cuculiza said that Fujimori's decline in health took her by surprise and that she had spoken with him five days earlier during which she noted his lucidity. Miguel Torres, a spokesperson for the Popular Force, added that Fujimori was going through a "difficult time". Fujimori's lawyer, Elio Riera, briefly disconnected from a virtual meeting over concerns for his health. Fujimori died at around 18:00 (UTC−05:00). A statement released by another doctor, José Carlos Gutiérrez, stated that Fujimori had trouble breathing on 9 September, lost consciousness on 10 September, and died from complications of tongue cancer. Keiko Fujimori later confirmed her father's death on social media. === State funeral and burial === The Peruvian government declared three days of mourning and granted him a state funeral. The Peruvian Congress and other public buildings lowered their flags to half-mast in his honor. Fujimori's remains were brought to lie in state at the Museo de la Nación in the Ministry of Culture on 12 September. Thousands of Fujimori supporters arrived from various regions of the country to the wake, carrying portraits and making speeches in his honor. Due to the large number of attendees, the Ministry of Culture announced that access to the wake would be extended until midnight, and that the following day, the doors of the Nasca Room would be open from 6 in the morning until midnight. From 12 to 14 September 2024, his lie in state at Peru's Ministry of Culture headquarters. Fujimori's state funeral was then held on 14 September 2024 at Lima's National Theatre. His funeral was attended by the incumbent president, Dina Boluarte, who offered a salute, and Keiko spoke during the funeral in front of a large portrait of her father. Fujimori was buried at Campo Fe Cemetery in Huachipa, Lima. === Reactions === President Dina Boluarte did not comment directly on his death, although her administration expressed its condolences to his family. Former presidents Francisco Sagasti, Manuel Merino, Martín Vizcarra and Pedro Pablo Kuczynski also conveyed their condolences, as well as prime minister Gustavo Adrianzén. Supporters also gathered at Fujimori's house to mourn his death. His death in his native Peru drew mixed reactions; congresssman Sigrid Bazán commented that Fujimori was a "dictator, assassin, and corrupt" and that "his legacy of corruptions, violations of human rights, and authoritarianism" would persist beyond his death. International media described him following his death as an "authoritarian" who was "divisive", and whose "heavy handed" tactics "created a negative legacy" in Peru that frustrated his eldest daughter's attempts to be elected to the presidency. Former president of Colombia Álvaro Uribe Vélez expressed his condolences and praised his administration, saying he "rescued Peru from many problems". Jamil Mahuad, former President of Ecuador, praised Fujimori and stated that he regretted "the loss of a friend". Yoshimasa Hayashi, Chief Cabinet Secretary of Japan, expressed his condolences to Fujimori's family, citing his role in resolving the Japanese embassy hostage crisis. At the same time, he acknowledged that Fujimori had been "evaluated in various ways" in part due to his human rights abuse cases. == Political actions == === Economic Policy === Fujimori’s tenure is perhaps best defined by his dramatic economic stabilization program, colloquially known as the "Fujishock", which produced significant, quantifiable improvements in Peru’s economic indicators. Prior to his reforms, the country suffered from hyperinflation that, at its peak, approached levels as high as 7,500% annually, while fiscal deficits were estimated to be in the range of 8–9% of GDP, and exports were roughly US$4 billion. Following the implementation of stringent fiscal and monetary policies, deregulation, and sweeping privatizations, inflation was slashed to single-digit levels (approximately 7–10% by 1994) and fiscal deficits were reduced to around 2–3% of GDP. In addition to these macroeconomic achievements, real GDP growth stabilized at an average of about 3–4% per annum during the mid-1990s, and export values soared from about US$4 billion in 1990 to over US$11 billion by 1997. One of the hallmarks of his administration was the privatization program, through which approximately 230–250 state-owned enterprises were sold, generating an estimated US$2.5–3 billion in capital inflows. This aggressive liberalization not only bolstered investor confidence but also helped increase foreign direct investment (FDI) to roughly 1.5–2% of GDP by the late 1990s. While these policies are widely credited with restoring macroeconomic stability and jumpstarting growth in a previously battered economy, they also contributed to heightened income inequality and social disparities. This is an enduring point of contention among economists and policy analysts. Critics argue that the rapid privatizations and deregulation, though successful in attracting capital, undermined public sector services and exacerbated regional imbalances. Nonetheless, the data support the conclusion that Fujimori’s economic interventions achieved a rapid and measurable turnaround in key economic indicators, laying the groundwork for subsequent decades of growth in Peru. === Corruption === Fujimori was accused of a series of offences, including embezzlement of public funds, abuse of power, and corruption during almost 10 years as president (1990–2000), especially when he gained greater control after the self-coup. The network operated as a kleptocracy in three spheres: business, politics, and the military. With multimillion-dollar annual expenditures in 1992 (five billion dollars in public spending plus another five billion in state enterprises), part of the funds were diverted to political and military institutions. According to the National Anti-Corruption Initiative (INA) in 2001, they corresponded to 30–35% of the average budget expenditure in each year, and 4% of the average annual GDP during the same period. One of those responsible for maintaining an image of apparent honesty and government approval was Vladimiro Montesinos, head of the National Intelligence Service (SIN), who systematically bribed politicians, judges, and the media. That criminal network also involved authorities of his government; furthermore, due to privatisation and the arrival of foreign capital, companies close to the Ministry of the Economy and Finance were allowed to use state money for public works tenders, as in the cases of AeroPerú, JJC Contratistas Generales (of the Camet Dickmann family), and the Banco de Crédito. Although in 1999 the opposition made a public denunciation that ended in the resignation of five ministers, this network was later revealed in 2000, just before the president resigned, when the Swiss embassy in Peru informed the Minister of Justice Alberto Bustamante and the attorney general José Ugaz of more than US$40 million coming from Montesinos, in which he was denounced for "illicit enrichment to the detriment of the Peruvian state". Ugaz was in charge of the investigation until 2002. According to Transparency International in 2004, Fujimori was listed as the seventh most corrupt former leader in history. ==== Counterterrorism efforts ==== When Fujimori came to power, much of Peru was dominated by the Maoist insurgent group Sendero Luminoso ("Shining Path"), and the Marxist–Leninist group Túpac Amaru Revolutionary Movement (MRTA). In 1989, 25% of Peru's district and provincial councils opted not to hold elections, owing to a persistent campaign of assassination, over the course of which over 100 officials had been killed by the Shining Path in that year alone. That same year, more than one-third of Peru's courts lacked a justice of the peace due to Shining Path intimidation. Labor union leaders and military officials were also assassinated throughout the 1980s. By the early 1990s, some parts of the country were under the control of the insurgents, in territories known as "zonas liberadas" ("liberated zones"), where inhabitants lived under the rule of these groups and paid them taxes. When the Shining Path arrived in Lima, it organized "paros armados" ("armed strikes"), which were enforced by killings and other forms of violence. The leadership of the Shining Path largely consisted of university students and teachers. Two previous governments, those of Fernando Belaúnde Terry and Alan García, at first neglected the threat posed by the Shining Path, then launched an unsuccessful military campaign to eradicate it, undermining public faith in the state and precipitating an exodus of elites. According to the Truth and Reconciliation Commission, Shining Path guerrilla attacks claimed an estimated 12,500 lives during the organization's active phase. On 16 July 1992, the Tarata bombing, in which several car bombs exploded in Miraflores, Lima's wealthiest district, killed over 40 people; the bombings were characterized by one commentator as an "offensive to challenge President Alberto Fujimori". The bombing at Tarata was followed up with a "weeklong wave of car bombings ... Bombs hit banks, hotels, schools, restaurants, police stations, and shops ... [G]uerrillas bombed two rail bridges from the Andes, cutting off some of Peru's largest copper mines from coastal ports." Fujimori earned credit for ending the Shining Path insurgency. As part of his anti-insurgency efforts, Fujimori granted the military broad powers to arrest suspected insurgents and try them in secret military courts with few legal rights. This measure has often been criticized for compromising the fundamental democratic and human right to an open trial wherein the accused faces the accuser. Fujimori contended that these measures were both justified and also necessary. Members of the judiciary were too afraid to charge the alleged insurgents, and judges and prosecutors had very legitimate fears of reprisals against them or their families. At the same time, Fujimori's government armed rural Peruvians, organizing them into groups known as rondas campesinas ("peasant patrols"). Insurgent activity was in decline by the end of 1992, and Fujimori took credit for this abatement, claiming that his campaign had largely eliminated the insurgent threat. After the 1992 auto-coup, the intelligence work of the DIRCOTE led to the capture of the leaders from↔ MRTA and the Shining Path, including notorious Shining Path leader Abimael Guzmán. Guzmán's capture was a political coup for Fujimori, who used it to great effect in the press; in an interview with documentarian Ellen Perry, Fujimori even noted that he specially ordered Guzmán's prison jumpsuit to be white with black stripes, to enhance the image of his capture in the media. Critics charge that to achieve the defeat of the Shining Path, the military engaged in widespread human rights abuses, and that the majority of the victims were poor highland countryside inhabitants caught in a crossfire between the military and insurgents. The final report of the Truth and Reconciliation Commission, published on 28 August 2003, noted that the armed forces were also guilty of destroying villages and murdering countryside inhabitants whom they suspected of supporting insurgents. The Japanese embassy hostage crisis began on 17 December 1996, when fourteen MRTA militants seized the residence of the Japanese ambassador in Lima during a party, taking hostage some four hundred diplomats, government officials, and other dignitaries. The action was partly in protest of prison conditions in Peru. During the four-month standoff, the Emerretistas gradually freed all but 72 of their hostages. The government rejected the militants' demand to release imprisoned MRTA members and secretly prepared an elaborate plan to storm the residence, while stalling by negotiating with the hostage-takers. On 22 April 1997, a team of military commandos, in the operation codenamed "Chavín de Huantar", raided the building. One hostage, two military commandos, and all 14 MRTA insurgents were killed in the operation. Images of President Fujimori at the ambassador's residence during and after the military operation, surrounded by soldiers and liberated dignitaries, and walking among the corpses of the insurgents, were widely televised. The conclusion of the four-month-long standoff was used by Fujimori and his supporters to bolster his image as tough on terrorism. ==== Human rights violations ==== Several organizations criticized Fujimori's methods against the Shining Path and the MRTA. Amnesty International said "the widespread and systematic nature of human rights violations committed during the government of former head of state Alberto Fujimori (1990–2000) in Peru constitute crimes against humanity under international law". The 1992 La Cantuta massacre and the 1991 Barrios Altos massacre by members of the Grupo Colina death squad, made up solely of members of the armed forces, were among the crimes that Peru cited in its request to Japan for his extradition in 2003. The success of the military operation in the Japanese embassy hostage crisis was tainted by subsequent allegations that at least three and possibly eight of the insurgents were summarily executed by the commandos after surrendering. In 2002, the case was taken up by public prosecutors, but the Supreme Court ruled that the military tribunals had jurisdiction. A military court later absolved them of guilt, and the Chavín de Huantar soldiers led the 2004 military parade. In 2003, MRTA family members lodged a complaint with the Inter-American Commission on Human Rights (IACHR) accusing the Peruvian state of human rights violations, in that the MRTA insurgents had been denied the "right to life, the right to judicial guarantees and the right to judicial protection". Although the IACHR's ruling did not directly implicate Fujimori, it did fault the Peruvian state for its complicity in the La Cantuta massacre. ==== Forced sterilizations ==== Reportedly following socioeconomic objectives calling for the "total extermination" of "culturally backward and economically impoverished groups" determined by the Peruvian military in the Plan Verde, from 1996 to 2000, the Fujimori government oversaw a massive forced sterilization campaign known as the National Program for Reproductive Health and Family Planning (PNSRPF). According to Back and Zavala, the plan was an example of ethnic cleansing as it targeted indigenous and rural women. The United Nations and other international aid agencies supported this campaign. USAID provided funding and training until it was exposed by objections by churches and human rights groups. The Nippon Foundation, headed by Ayako Sono, a Japanese novelist and personal friend of Fujimori, supported it as well. In the four-year Plan Verde period, over 215,000 people, mostly women, entirely indigenous, were forced or threatened into sterilization and 16,547 men were forced to undergo vasectomies during these years, most of them without a proper anesthetist, in contrast to 80,385 sterilizations and 2,795 vasectomies over the previous three years. Some scholars argue that these policies and acts were genocidal. == Legacy == === Economics === Fujimori was credited by many Peruvians with bringing stability to the country after the violence and hyperinflation during the first García administration. Neoliberal reforms under Fujimori took place in three distinct phases: an initial "orthodox" phase (1990–92) in which technocrats dominated the reform agenda; a "pragmatic" phase (1993–98) that saw the growing influence of business elites over government priorities; and a final "watered-down" phase (1999–2000) dominated by a clique of personal loyalists and their clientelist policies that aimed to secure Fujimori a third term as president. Business was a big winner of the reforms, with its influence increasing significantly within both the state and society. High growth during Fujimori's first term petered out during his second term. The 1997–98 El Niño event had a tremendous impact on the Peruvian economy during the late 1990s and exacerbated a recession during that time. Nevertheless, total GDP growth between 1992 and 2001, inclusive, was 44.60%, that is, 3.76% per annum; total GDP per capita growth between 1991 and 2001, inclusive, was 30.78%, that is, 2.47% per annum. Also, studies by INEI, the national statistics bureau show that the number of Peruvians living in poverty increased dramatically (from 41.6% to more than 70%) during Alan García's term, but decreased greatly (from more than 70% to 54%) during Fujimori's term. Furthermore, FAO reported Peru reduced undernourishment by about 29% from 1990 to 1992 to 1997–99. Peru was reintegrated into the global economic system, and began to attract foreign investment. The mass selloff of state-owned enterprises led to improvements in some service industries, notably local telephone, mobile telephone, and internet services, respectively. For example, before privatization, a consumer or business had to wait up to 10 years to get a local telephone line installed by the state-run telephone company at a cost of US$607 for a residential line. Peru's physical land-based telephone network had a dramatic increase in telephone penetration from 2.9% in 1993 to 5.9% in 1996 and 6.2% in 2000, and a dramatic decrease in the wait for a telephone line: the average wait went from 70 months in 1993 (before privatization) to two months in 1996 (after privatization). Privatization also generated foreign investment in export-oriented activities such as mining and energy extraction, notably the Camisea Gas Project and the copper and zinc extraction projects at Antamina. === Criticism === Fujimori has been described as a dictator. His government was permeated by a network of corruption organized by his associate Montesinos. Fujimori's style of government has also been described as "populist authoritarianism". Numerous governments and human rights organizations such as Amnesty International, welcomed the extradition of Fujimori to face human rights charges. As early as 1991, Fujimori had himself vocally denounced what he called "pseudo-human rights organizations" such as Amnesty International and Human Rights Watch, for allegedly failing to criticize the insurgencies targeting civilian populations throughout Peru against which his government was struggling. Some of the GDP growth during the Fujimori years actually reflects a greater rate of extraction of nonrenewable resources by transnational companies; these companies were attracted by Fujimori by means of near-zero royalties, and, by the same fact, little of the extracted wealth has stayed in the country. Peru's mining legislation, they claim, has served as a role model for other countries that wish to become more mining-friendly. The sole instance of organized labor's success in impeding reforms, namely the teachers' union resistance to education reform, was based on traditional methods of organization and resistance: strikes and street demonstrations. In the 2004 Global Corruption Report, Fujimori was named one of the World's Most Corrupt Leaders. He was listed seventh and he was said to have amassed $600 million, but despite years of incarceration and investigation, none of these supposed stolen funds have ever been located in any bank account anywhere in the world. === Support === Fujimori did have support within Peru. The Universidad de Lima March 2003 poll, taken while he was in Japan, found a 41% approval rating for his administration. A poll conducted in March 2005 by the Instituto de Desarrollo e Investigación de Ciencias Económicas (IDICE) indicated that 12.1% of the respondents intended to vote for Fujimori in the 2006 presidential election. A poll conducted on 25 November 2005, by the Universidad de Lima indicated a high approval (45.6%) rating of the Fujimori period between 1990 and 2000, attributed to his counterinsurgency efforts (53%). In a 2007 University of Lima survey of 600 Peruvians in Lima and the port of Callao, 82.6% agreed that the former president should be extradited from Chile to stand trial in Peru. In the 2006 congressional elections, his daughter Keiko was elected to the congress with the highest vote count. She came in second place in the 2011 Peruvian presidential election with 23.2% of the vote, and lost the June runoff against Ollanta Humala. She again ran for president in the 2016 election, narrowly losing the runoff to Pedro Pablo Kuczynski, and again in the 2021 election, losing the runoff to Pedro Castillo. === Honors === Mexico: Collar of the Order of the Aztec Eagle (1996) Thailand: Grand Cordon (Special Class) of the Order of the White Elephant (1996) Austria: Grand Star of the Decoration of Honour for Services to the Republic of Austria (1997) United Kingdom: Honorary Knight Grand Cross of the Order of St Michael and St George (1998) Brazil: Grand Collar of the Order of the Southern Cross (1999) Malaysia: Honorary Recipient of the Order of the Crown of the Realm (1996) == See also == == Notes == == References == == External links == Biography and tenure by CIDOB Foundation (in Spanish) Appearances on C-SPAN Alberto Fujimori on Charlie Rose Alberto Fujimori at IMDb Alberto Fujimori collected news and commentary at The New York Times The Fall of Fujimori on POV at PBS, 2006 State of Fear a documentary of Peru's war on terror based on the findings of the Peruvian Truth and Reconciliation Commission
Wikipedia:Alberto González Domínguez#0	Alberto González Domínguez (11 April 1904 in Buenos Aires – 14 September 1982 in Buenos Aires) was an Argentine mathematician working on analysis, probability theory and quantum field theory. González Domínguez received his Ph.D. from the University of Buenos Aires in 1939 under the direction of Julio Rey Pastor. That same year, González Domínguez received a Guggenheim Fellowship and worked for two years with Jacob Tamarkin at Brown University. González Domínguez spent most of his career as a professor at the University of Buenos Aires. == References == == External links == Biographical Sketch Full curriculum vitae including the list of publications Alberto González Domínguez at the Mathematics Genealogy Project
Wikipedia:Alberto Tognoli#0	Alberto Tognoli (born 26 July 1937, Brescia, died 3 March 2008 in Rapallo) was an Italian mathematician, who worked on algebraic geometry. Tognoli received his Ph.D. (Laurea) in 1960 from the University of Pisa. From 1970 he became full professor at the same university, and he also taught in Cosenza, Ferrara, Paris and Tours. He was also a professor of geometry at the University of Trento from 1986 until his retirement as professor emeritus in 2005. He is known for his research on Nash functions and Nash manifolds (named after John Nash). He proved Nash's conjecture that smooth compact manifolds are diffeomorphic to non-singular real algebraic manifolds (known as the Theorem of Nash and Tognoli). He received in 1974 the Caccioppoli Prize and in 1988 the Gold Medal in Mathematics of the Accademia dei XL. Selected publications Algebraic Geometry and Nash Functions, Institutiones Mathematicae, Academic Press 1978 as editor with M. Galbiati: Real analytic and algebraic geometry : proceedings of the conference held in Trento, Italy, October 3–7, 1988, Lecture Notes in Mathematics 1420, Springer Verlag 1990 Singularities of Analytic Spaces, CIME, Rom: Cremonese 1975 Introduzione alla teoria degli spazi analitici reali, Rom, Accademia dei Lincei 1976 Approximation theorems and Nash conjecture, Memoires SMF, 38, 1974, 53-68, numdam Algebraic approximation of manifolds and spaces, Seminaire Bourbaki, No. 548, 1979/80, numdam Tognoli, Alberto (1988). "Analyticity of homology classes". Proc. Amer. Math. Soc. 104 (3): 920–922. doi:10.1090/s0002-9939-1988-0964874-2. MR 0964874. with Alessandro Tancredi: Tancredi, Alessandro; Tognoli, Alberto (2006). "On the products of Nash subvarieties by spheres". Proc. Amer. Math. Soc. 134 (4): 983–987. doi:10.1090/S0002-9939-05-08246-8. MR 2196028. == References == == External links == Produ zione Scientifica – Alberto Tognoli
Wikipedia:Aleksandar Nikolov (computer scientist)#0	Aleksandar Nikolov is a Bulgarian and Canadian theoretical computer scientist working on differential privacy, discrepancy theory, and high-dimensional geometry. He is a professor at the University of Toronto. Nikolov obtained his Ph.D. from Rutgers University in 2014 under the supervision of S. Muthukrishnan (Thesis: New computational aspects of discrepancy theory). Nikolov is the Canada Research Chair in Algorithms and Private Data Analysis. == Early life and education == Aleksandar Nikolov was born in Varna, Bulgaria, to Bulgarian parents of Banat origin. After attaining primary and secondary education in Varna, he was awarded a presidential scholarship to St. Peter's University in Jersey City, NJ. There he majored in computer science, graduating as the valedictorian of his class. He later pursued graduate studies at Rutgers University, obtaining his PhD in computer science under the supervision of S. Muthukrishnan. == Academic and research career == Following his doctoral studies, Nikolov joined the University of Toronto as an assistant professor in the Department of Computer Science. His research interests include differential privacy, optimization, and the design and analysis of algorithms. Nikolov has made significant contributions to understanding the mathematical foundations of privacy and the development of efficient algorithms with strong theoretical guarantees. == Research contributions == Nikolov's work on differential privacy has been particularly influential. He has explored various aspects of privacy-preserving data analysis, including mechanisms for ensuring privacy in statistical queries and optimization problems. His research has advanced the theoretical understanding of how to balance data utility with privacy guarantees. In addition to privacy, Nikolov has contributed to the field of optimization, focusing on algorithms for high-dimensional data and the development of efficient approximation algorithms. His work often intersects with machine learning, where he addresses problems related to data representation and complexity. == Publications and citations == Aleksandar Nikolov has published numerous papers in prestigious conferences and journals. His work has been widely cited, reflecting the impact of his research on the computer science community. Some of his notable publications include contributions to conferences such as STOC (Symposium on Theory of Computing), FOCS (Foundations of Computer Science), and SODA (Symposium on Discrete Algorithms). == Teaching and mentorship == At the University of Toronto, Nikolov is also recognized for his dedication to teaching and mentorship. He teaches courses on algorithms, data privacy, and theoretical computer science, inspiring a new generation of computer scientists. His approach to teaching emphasizes both the theoretical underpinnings of computer science and their practical applications. == Professional activities and recognition == Nikolov is actively involved in the academic community, serving on program committees for major conferences and reviewing for top journals. His contributions have earned him recognition, highlighting his role as a leading figure in his areas of expertise. == Recent work and future directions == In recent years, Nikolov has focused on developing a 21st-century framework for addressing contemporary problems in computer science. He advocates for interdisciplinary approaches that combine insights from computer science, mathematics, and other fields to tackle complex challenges. His recent initiatives aim to bridge the gap between theoretical research and practical applications, ensuring that advancements in algorithms and privacy have a meaningful impact on society. == External links == Aleksandar Nikolov's Google Scholar Profile Aleksandar Nikolov's Research Page at the University of Toronto Aleksandar Nikolov's Personal Homepage at the University of Toronto Article on the Need for a 21st-Century Framework for 21st-Century Problems == References ==
Wikipedia:Aleksander Axer#0	Aleksander Axer (10 October 1880 – 4 October 1948) was a Polish mathematician from Przemyśl who introduced Axer's theorem. He died suddenly of complications from pharyngitis in Zurich, shortly before his 68th birthday. == References ==
Wikipedia:Aleksander Rajchman#0	Aleksander Michał Rajchman (13 November 1890 – July or August 1940) was a mathematician of the Warsaw School of Mathematics of the Interwar period. He had origins in the Lwów School of Mathematics and contributed to real analysis, probability and mathematical statistics. == Family background == Rajchman was born on 13 November 1890 in Warsaw, Congress Poland, a province of the Russian Empire, in the family of assimilated Polish Jews known for contributions to the 20th-century Polish intellectual life. Although the family was partially converted into Roman Catholicism, his parents were agnostic. His father Aleksander Rajchman was a journalist specialized in theatre and music critique, who in the period 1882-1904 was the publisher and editor-in-chief of the artistic weekly Echo Muzyczne, Teatralne i Artystyczne and was co-founder and first director of the National Philharmonic in Warsaw in the years 1901–1904. Mother Melania Amelia Hirszfeld was a socialist and women's rights activist who wrote both critical essays and woman affairs' texts under pseudonyms or anonymously for a few Polish weeklies, organized maternal rallies where she drew attention to the need to improve the household to facilitate women's lives, and was an active member of the secret organization Women's Circle of Polish Crown and Lithuania, and later also the Association of Women's Equality in Warsaw. Rajchmans ran a social salon who hosted many Polish artists of their times, in particular Eliza Orzeszkowa, Maria Konopnicka, and Zenon Pietkiewicz. His older sister a Polish independence activist and historian of education Helena Radlińska was the founder of Polish social pedagogy, his older brother a physician and bacteriologist Ludwik Rajchman was the world leader in social medicine and director of the League of Nations Health Organization, the founder of the United Nations International Children's Emergency Fund (UNICEF) and its first chairman in the years 1946–1950. His nephew a Polish-American electrical engineer Jan A. Rajchman was a computer pioneer who invented logic circuits for arithmetic and magnetic-core memory to result in development of high-speed computer memory systems and whose son John Rajchman is a noted American philosopher of art history, architecture, and continental philosophy. His first cousin a microbiologist and serologist Ludwik Hirszfeld co-discovered the heritability of ABO blood group type and foreseen the serological conflict between mother and child. == Education and Research Work == After his father died in 1904, his mother migrated with rest of the family to Paris in 1909. Alexander studied there and obtained the licencié és sciences degree in 1910. He became a junior assistant at the University of Warsaw in 1919, whereas in 1921 he earned the doctoral degree at the John Casimir University of Lwów under Hugo Steinhaus and became a senior assistant at the University of Warsaw. Next in 1922 he became a professor at the University of Warsaw, and, after his habilitation in 1925, a lecturer there until outbreak of the World War II in 1939. In the 1930s, he was a visiting scholar to lecture at the Jacques Hadamard's seminar at the Collège de France. His research touched real analysis, probability and mathematical statistics, in particular focused on the Fourier series. Rajchman received significant results in the fields of trigonometric series, function of a real variable and probability. In mathematics, there are such concepts as the Rajchman global uniqueness theorem, Rajchman measures, Rajchman collection, Rajchman algebras, Rajchman sharpened law of large numbers, Rajchman theory of formal multiplication of trigonometric series, Rajchman inequalities, and Rajchman-Zygmund inequalities. Near a Rajchman measure, particularly important notion invented by Rajchman is a Rajchman algebra associated with a locally compact group which is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity, a closed and complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. His first doctoral student a noted Polish mathematician Antoni Zygmund created the Chicago school of mathematical analysis with the emphasis onto harmonic analysis, which produced the 1966 Fields Medal winner Paul Cohen. His second doctoral student Zygmunt Zalcwasser, co-advised by Wacław Sierpiński, introduced the Zalcwasser rank to measure the uniform convergence of sequences of continuous functions on the unit interval. In October 2000, the Stefan Banach International Mathematical Center at the Institute of Mathematics of the Polish Academy of Sciences honoured Rajchman's achievements by the Rajchman-Zygmund-Marcinkiewicz Symposium. In April 1940, the Gestapo arrested Rajchman as a Jew. He died in Sachsenhausen concentration camp, Oranienburg, Germany probably in July or August 1940. == See also == List of Polish mathematicians == References == Aleksander Rajchman at the Mathematics Genealogy Project Aleksander Rajchman from an article by Antoni Zygmund Zygmund, Antoni (1987), "Aleksander Rajchman (1890–1940)", Roczniki Polskiego Towarzystwa Matematycznego. Seria II. Wiadomości Matematyczne (in Polish), 27 (2): 219–231, ISSN 0373-8302, MR 0908884 Maligranda, Lech; Piotrowski, Walerian (2017), "Aleksander Rajchman (1890–1940)" (PDF), Roczniki Polskiego Towarzystwa Matematycznego. Seria II. Wiadomości Matematyczne (in Polish), 53 (1): 43–82
Wikipedia:Aleksandr Aleksandrov (mathematician)#0	Aleksandr Danilovich Aleksandrov (Russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: Alexandr or Alexander (first name), and Alexandrov (last name)) (4 August 1912 – 27 July 1999) was a Soviet/Russian mathematician, physicist, philosopher and mountaineer. == Personal life == Aleksandr Aleksandrov was born in 1912 in Volyn, Ryazan Oblast. His father was a headmaster of a secondary school in St Petersburg and his mother a teacher at said school, thus the young Alekandrov spent a majority of his childhood in the city. His family was old Russian nobility—students noted ancestral portraits which hung in his office. His sisters were Soviet botanist Vera Danilovna Aleksandrov (RU) and Maria Danilovna Aleksandrova, author of the first monograph on gerontopsychology in the USSR. In 1937, he married a student of the Faculty of Physics, Marianna Leonidovna Georg. Together they had two children: Daria (b. 1948) and Daniil (RU) (b. 1957). In 1980, he married Svetlana Mikhailovna Vladimirova (nee Bogacheva). In 1951 he became a member of the Communist Party. Alekandrov had a personal love for poetry, writing and translating. Once, on a trip to London, he was received as a visiting Shakespeare scholar. He was also very well travelled, visiting India, the US, and throughout Europe. == Scientific career == He graduated from the Department of Physics of Leningrad State University. His advisors there were Vladimir Fock, a physicist, and Boris Delaunay, a mathematician. In 1933 Aleksandrov worked at the State Optical Institute (GOI) and at the same time gave lectures at the Department of Mathematics and Mechanics of the university. He completed his Ph.D. in 1935 at the university and later in 1937 — a D.Sc. dissertation. He became a professor at the university, while also working at LOMI, the Leningrad Department of the Steklov Mathematical Institute (now PDMI, Petersburg Department of the Mathematical Institute). Appointed the rector of the university in 1952, Aleksandrov remained in this position until 1964. He was the youngest rector in university history, and was fairly popular. One of his main contributions was the attempted move of Leningrad State University to Old Peterhof, which proved unsuccessful. In 1946 he became a corresponding member, and in 1964 a full member of the USSR Academy of Sciences. Since1975 he was also a member of the Accademia dei Quaranta. From 1964 to 1986 Aleksandrov lived in Novosibirsk, heading the Laboratory of Geometry of the Institute of Mathematics of the Siberian Division of the USSR Academy of Sciences, teaching at Novosibirsk State University. In 1986 he returned to Leningrad (now Saint Petersburg) to head the geometry laboratory at LOMI. Aleksandrov's main work was in the study of differential geometry and physics. His work in geometry specifically is said to be second only to Gauss by N. V. Efimov, V. A. Zalgaller and A. V. Pogorelov. == Awards == Partial list of the awards, medals, and prizes awarded to Aleksandrov: Stalin Prize (1942) Lobachevsky International Prize (1951) Euler Gold Medal of the Russian Academy of Sciences (1992) One of the many orders that he was awarded was given to him in 1990 for his efforts in preserving genetics from the attacks of the pseudoscience of Lysenkoism that had official state support in the times of Stalin and Khrushchev. == Works by Aleksandrov == Aleksandrov wrote a multitude of books, scientific papers, textbooks for various levels (schools to universities), including Convex Polyhedra, originally published in Russian in 1950 and translated into English in 2005. He also wrote non-mathematical papers, memoirs about famous scientists, and philosophical essays dealing with the moral values of science. A full bibliography is available in [1]. Selected works are available in English: Alexandrov, A.D. Selected works. Part 1: Selected scientific papers. Amsterdam: Gordon and Breach Publishers. x, 322 p. (1996). ISBN 2-88124-984-1 Alexandrov, A.D. Selected works. Intrinsic geometry of convex surfaces. Vol. 2. Boca Raton, FL: Chapman & Hall/CRC. xiii, 426 p. (2005). ISBN 0-415-29802-4 Alexandrov, A.D. Convex polyhedra. Springer: Berlin. xi, 539 p. (2005). ISBN 3-540-23158-7 (1st edition, 1950) Alexandrov, A.D. Die innere Geometrie der konvexen Flächen. Akademie Verlag. (1955). (German translation of 1948 Russian original) == Students of Aleksandrov == I. Liberman, S. Olovianishnikoff, P. Kostelyanetz — all the three of them died on the battlefields of World War II A. Pogorelov — from Kharkov A. Yusupov — from Bukhara Students from the Aleksandrov Leningrad period (ordered by the time of joining the seminars): Yu. Borisov, V. Zalgaller, Yu. Reshetnyak, I. Bakelman, Yu. Volkov, A. Zamorzaev, S. Bogacheva (who later married Aleksandrov), Yu. Borovskii, R. Pimenov Sobchuk and Starokhozyayev — from Ukraine G. Rusiyeshvili — from Georgia (country) B. Frank and H. Frank — from Germany Yu. Burago, V. Kreinovich; Grigori Perelman Moved from Alma-Ata after Aleksandrov's lecture tour there: M. Kvachko, V. Ovchinnikova, E. Sen'kin Stayed in Alma-Ata: A. Zilberberg, V. Strel'cov, D. Yusupov Novosibirsk students: A. Guts, A. Kuz'minykh, A. Levichev, and A. Shaidenko. Both in St. Petersburg and Novosibirsk Aleksandrov participated in joint research also with some of his students' students. Several of them became his co-authors: V. Berestovskii, A. Verner, N. Netsvetaev, I. Nikolaev, and V. Ryzhik. His last Ph.D. student was Grigori Perelman who proved Thurston's geometrization conjecture in 2002/2003 which contains the Poincaré conjecture as a special case. == Mountaineering == Aleksandrov became attracted to alpinism under the influence of his advisor Boris Delaunay. In the summer of 1937, after defending his D.Sc., …together with I. Chashnikov he makes a first climb to the Chotchi summit, and with K. Piskaryov performs a climb of Bu-Ul'gen via the western wall (one of the first wall climbs in the history of the Soviet alpinism).[…] In 1940 he participates in a record-making traversal[…] He manages, almost by a miracle, to stop the fall of A. Gromov, who had fallen along with a snow shelf. It was with this traversal that Aleksandrov completed the alpinist sports master requirements. The German-Soviet War postponed awarding him this honorary title until 1949. During his rectorship, Aleksandrov also advanced the mountaineering sport activities in the university, actively participating in the climbs. The fiftieth birthday was celebrated by Aleksandrov in the mountains with his friends. On that day he made a solo first climb of an …unnamed peak 6222 m (Shakhdarinsk ridge, Pamir), that as he suggested was then named "The peak of the Leningrad university." During later years Aleksandrov was unable to climb due to health problems, yet he never ceased dreaming of climbing. Finally, in 1982, the year of his seventieth birthday, he, together with K. Tolstov, performed in Tian Shan his last climb, of the Panfilov Peak… == External links == == See also == CAT(k) space Cauchy's theorem Alexandrov theorem Aleksandrov–Rassias problem Alexandrov–Fenchel inequality Alexandrov's uniqueness theorem Alexandrov’s soap bubble theorem == References == Академик Александр Данилович Александров. Воспоминания. Публикации. Материалы. (Academician Aleksandr Danilovich Aleksandrov. Recollections. Publications. Biographical materials, in Russian). Editors: G.M. Idlis and O.A. Ladyzhenskaya. Moscow, Nauka publishing house, 2002. Yu. F. Borisov, "On the 90th anniversary of the birth of A.D. Aleksandrov (1912–1999) Archived 2022-03-28 at the Wayback Machine", Russ. Math. Surv., 2002, 57 (5), 1017–1031. Yu. F. Borisov, V.A. Zalgaller, Kutateladze, S.S., O.A. Ladyzhenskaya, A.V. Pogorelov, Yu. G. Reshetnyak, "К 90-летию со дня рождения А.Д. Александрова (1912–1999)", Uspekhi Mat. Nauk, 2002, 57 (5), 169–181. Liyun Tan and Shuhuang Xiang, On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem, Banach Journal of Mathematical Analysis, 1(1)(2007), 11–22. A.M. Vershik, "Alexander Danilovich as I knew him (in Russian).", St. Petersburg University, No. 3-4 (2004), 36–40. Shuhuang, Xiang, On the Aleksandrov-Rassias problem for isometric mappings, Functional Equations, Inequalities and Applications, Kluwer Acad. Publ., Dordrecht, 2003, pp. 191–221. == External links == O'Connor, John J.; Robertson, Edmund F., "Aleksandr Aleksandrov (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews (with additional photos) Aleksandr Aleksandrov at the Mathematics Genealogy Project (incomplete students listing as of December 2004) Review of Alexandrov's "Convex Polytopes" – by R. Connelly, published at the Mathematical Reviews. Alexandr Danilovich Alexandrov – biography, reminiscences, references (from St. Petersburg Mathematical Society website) Traits – by S.S. Kutateladze Alexandrov Par Excellence – by S.S. Kutateladze Alexandrov of Ancient Hellas – by S.S. Kutateladze Author profile in the database zbMATH The Rector (a film)
Wikipedia:Aleksandr Korkin#0	Aleksandr Nikolayevich Korkin (Russian: Александр Николаевич Коркин; 3 March [O.S. 19 February] 1837 – 1 September [O.S. 19 August] 1908) was a Russian mathematician. He made contribution to the development of partial differential equations, and was second only to Chebyshev among the founders of the Saint Petersburg Mathematical School. Among others, his students included Yegor Ivanovich Zolotarev. == Some publications == Korkine A., Zolotareff G. (1872). "Sur les formes quadratiques positives quaternaires". Math. Ann. 5 (4): 581–583. doi:10.1007/BF01442912. S2CID 119606974. Korkine A., Zolotareff G. (1873). "Sur les formes quadratiques". Math. Ann. 6 (3): 366–389. doi:10.1007/BF01442795. S2CID 120492026. Korkine A., Zolotareff G. (1877). "Sur les formes quadratiques positives". Math. Ann. 11 (2): 242–292. doi:10.1007/BF01442667. S2CID 121803621. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Aleksandr Korkin", MacTutor History of Mathematics Archive, University of St Andrews Aleksandr Korkin at the Mathematics Genealogy Project Korkin's Biography Archived 2014-12-19 at the Wayback Machine, the St. Petersburg University Pages (in Russian, but with an image)
Wikipedia:Aleksandr Kurosh#0	Aleksandr Gennadyevich Kurosh (Russian: Алекса́ндр Генна́диевич Ку́рош; January 19, 1908 – May 18, 1971) was a Soviet mathematician, known for his work in abstract algebra. He is credited with writing The Theory of Groups, the first modern and high-level text on group theory, published in 1944. He was born in Yartsevo, in the Dukhovshchinsky Uyezd of the Smolensk Governorate of the Russian Empire and died in Moscow. He received his doctorate from the Moscow State University in 1936 under the direction of Pavel Alexandrov. In 1937 he became a professor there, and from 1949 until his death he held the Chair of Higher Algebra at Moscow State University. In 1938, he was the PhD thesis adviser to his fellow group theory scholar Sergei Chernikov, with whom he would develop important relationships between finite and infinite groups, discover the Kurosh-Chernikov class of groups, and publish several influential papers over the next decades. In all, he had 27 PhD students, including also Vladimir Andrunakievich, Mark Graev, and Anatoly Shirshov. On the whole stretch of a long and very fruitful period 1930–1971, A. G. Kurosh and his students have obtained many interesting and deep results in the theory of associative algebras, lattice theory, general theory of radicals, theory of categories, theory of universal algebras, linear multioperator rings and algebras, Ω-rings, etc. == Selected publications == Teoriya Grupp (Теория групп), 2 vols., Nauk, 1944, 2nd edition 1953. German translation: Gruppentheorie. 2 vols., 1953, 1956, Akademie Verlag, Berlin, 2nd edition 1970, 1972. English translation: The Theory of Groups, 2 vols., Chelsea Publishing Company, the Bronx, tr. by K. A. Hirsch, 1950, 2nd edition 1955. Vorlesungen über Allgemeine Algebra. Verlag Harri Deutsch, Zürich 1964. Zur Theorie der Kategorien. Deutscher Verlag der Wissenschaften, Berlin 1963. Kurosch: Zur Zerlegung unendlicher Gruppen. Mathematische Annalen vol. 106, 1932. Kurosch: Über freie Produkte von Gruppen. Mathematische Annalen vol. 108, 1933. Kurosch: Die Untergruppen der freien Produkte von beliebigen Gruppen. Mathematische Annalen, vol. 109, 1934. A. G. Kurosh, S. N. Chernikov, “Solvable and nilpotent groups”, Uspekhi Mat. Nauk, 2:3(19) (1947), 18–59. A. G. Kurosh, "Curso de Álgebra Superior", Editorial Mir, Moscú 1997, traducción de Emiliano Aparicio Bernardo (in Spanish) == See also == Kurosh subgroup theorem Kurosh problem == References == == External links == Aleksandr Kurosh at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Aleksandr Kurosh", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Aleksandr Lyapunov#0	Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, 6 June [O.S. 25 May] 1857 – 3 November 1918) was a Russian mathematician, mechanician and physicist. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory. == Biography == === Early life === Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer employed by the Demidov Lyceum. His brother, Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (now Ulyanovsk Oblast). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of the physiologist Ivan Mikhailovich Sechenov. At his uncle's family, Lyapunov studied with his distant cousin Natalia Rafailovna, who became his wife in 1886. In 1870, his mother moved with her sons to Nizhny Novgorod, where he started the third class of the gymnasium. He graduated from the gymnasium with distinction in 1876. === Education === In 1876, Lyapunov entered the Physico-Mathematical department at the University of Saint Petersburg, but after one month he transferred to the Mathematics department of the university. Among the Saint Petersburg mathematics professors were Chebyshev and his students Aleksandr Nikolaevich Korkin and Yegor Ivanovich Zolotarev. Lyapunov wrote his first independent scientific works under the guidance of the professor of mechanics, D. K. Bobylev. In 1880 Lyapunov received a gold medal for a work on hydrostatics. This was the basis for his first published scientific works On the equilibrium of a heavy body in a heavy fluid contained in a vessel of a fixed form and On the potential of hydrostatic pressure. Lyapunov also completed his university course in 1880, two years after Andrey Markov who had also graduated at Saint Petersburg University. Lyapunov maintained scientific contact with Markov throughout his life. === Teaching and research === A major theme in Lyapunov's research was the stability of a rotating fluid mass with possible astronomical application. This subject was proposed to Lyapunov by Chebyshev as a topic for his masters thesis which he submitted in 1884 with the title On the stability of ellipsoidal forms of rotating fluids. The main contribution was published in the celebrated monograph 'A.M. Lyapunov, The general problem of the stability of motion. 1892. Kharkov Mathematical Society, Kharkov, 251p. (in Russian)'. This led on to his 1892 doctoral thesis The general problem of the stability of motion. The thesis was defended in Moscow University on 12 September 1892, with Nikolai Zhukovsky and V. B. Mlodzeevski as opponents. In 1908, the Kharkov edition was translated to French and republished by the University of Toulouse: 'Probleme General de la Stabilite du Mouvement, Par M.A. Liapounoff. Traduit du russe par M.Edouard Davaux'. In 1885, Lyapunov became privatdozent and was proposed to accept the chair of mechanics at Kharkov University, where he went the same year. About the initial stay at Kharkov, Smirnov writes in his biography of Lyapunov: Here at first, the research activity of Lyapunov was cut short. It was necessary to work out courses and put together notes for students, which took up much time. His student and collaborator, Vladimir Steklov, recalled his first lecture in the following way: "A handsome young man, almost of the age of the other students, came before the audience, where there was also the old Dean, professor Levakovsky, who was respected by all students. After the Dean had left, the young man with a trembled voice started to lecture a course on the dynamics of material points, instead of a course on dynamical systems. This subject was already known to the students from the lectures of professor Delarue. But what Lyapunov taught us was new to me and I had never seen this material in any textbook. All antipathy to the course was immediately blown to dust. From that day students would show Lyapunov a special respect." === Later years === Lyapunov returned to Saint Petersburg in 1902, after being elected acting member of the Academy of Science as well as ordinary professor in the Faculty of Applied Mathematics of the university. The position had been left vacant by the death of his former teacher, Chebyshev. Not having any teaching obligations, this allowed Lyapunov to focus on his studies and in particular he was able to bring to a conclusion the work on the problem of Chebyshev with which he started his scientific career. In 1908, he took part to the Fourth International Mathematical Congress in Rome. He also participated in the publication of Euler's selected works: he was an editor of the volumes 18 and 19. === Death === By the end of June 1917, Lyapunov traveled with his wife to his brother's palace in Odessa. Lyapunov's wife was suffering from tuberculosis so they moved in accordance with her doctor's orders. She died on 31 October 1918. The same day, Lyapunov shot himself in the head, and three days later he died. By that time, he was going blind from cataracts. == Work == Lyapunov contributed to several fields, including differential equations, potential theory, dynamical systems and probability theory. His main preoccupations were the stability of equilibria and the motion of mechanical systems, especially rotating fluid masses, and the study of particles under the influence of gravity. His work in the field of mathematical physics regarded the boundary value problem of the equation of Laplace. In the theory of potential, his work from 1897 On some questions connected with Dirichlet's problem clarified several important aspects of the theory. His work in this field is in close connection with the work of Steklov. Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. In the theory of probability, he generalized the works of Chebyshev and Markov, and proved the Central Limit Theorem under more general conditions than his predecessors. The method of characteristic functions he used for the proof later found widespread use in probability theory. Like many mathematicians, Lyapunov preferred to work alone and communicated mainly with few colleagues and close relatives. He usually worked late, four to five hours at night, sometimes the whole night. Once or twice a year he visited the theatre, or went to some concert. He had many students. He was an honorary member of many universities, an honorary member of the academy in Rome and a corresponding member of the Academy of Sciences in Paris. Lyapunov's impact was significant, and the following mathematical concepts are named after him: Lyapunov equation Lyapunov exponent Lyapunov function Lyapunov fractal Lyapunov stability Lyapunov's central limit theorem Lyapunov vector == Selected publications == 1884, On the stability of ellipsoidal figures of equilibrium of a rotating fluid (in Russian) Published in Bulletin Astronomique 1885 1892, A.M. Lyapunov, The general problem of the stability of motion. 1892. Kharkov Mathematical Society, Kharkov, 251p. (in Russian) 1897, Sur certaines questions qui se rattachent au problème de Dirichlet 1901, Nouvelle forme du théorème sur la limite de probabilité 1901, Sur un théorème du calcul des probabilités 1902, Sur une série dans la théorie des équations différentielles linéaires du second ordre à coefficients périodiques 1903, Recherches dans la théorie de la figure des corps célestes 1904, Sur l'équation de Clairaut et les équations plus générales de la théorie de la figure des planètes == See also == == Notes == == References == == Further reading == == External links == O'Connor, John J.; Robertson, Edmund F., "Aleksandr Lyapunov", MacTutor History of Mathematics Archive, University of St Andrews Aleksandr Lyapunov at the Mathematics Genealogy Project Ляпунов Александр Михайлович at www. mathsoc.spb. ru (in Russian) Ляпунов Александр Михайлович (1857-1918) Archived 2007-02-23 at the Wayback Machine at www.spbu. ru (in Russian) Ляпунов Александр Михайлович at www-mechmath. univer. kharkov. ua (in Russian) [1] Aleksandr M. Lyapunov = Ляпунов Александр Михайлович alive at scholar.google.com (live citations)
Wikipedia:Aleksandrov–Rassias problem#0	The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932. They proved the Mazur–Ulam theorem stating that every isometry of a normed real linear space onto a normed real linear space is a linear mapping up to translation. In 1970, Aleksandr Danilovich Aleksandrov asked whether the existence of a single distance that is preserved by a mapping implies that it is an isometry, as it does for Euclidean spaces by the Beckman–Quarles theorem. Themistocles M. Rassias posed the following problem: Aleksandrov–Rassias Problem. If X and Y are normed linear spaces and if T : X → Y is a continuous and/or surjective mapping such that whenever vectors x and y in X satisfy ‖ x − y ‖ = 1 {\displaystyle \lVert x-y\rVert =1} , then ‖ T ( X ) − T ( Y ) ‖ = 1 {\displaystyle \lVert T(X)-T(Y)\rVert =1} (the distance one preserving property or DOPP), is T then necessarily an isometry? There have been several attempts in the mathematical literature by a number of researchers for the solution to this problem. == References == P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (eds.), Nonlinear Analysis. Stability, Approximation, and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday, Springer, New York, 2012. A. D. Aleksandrov, Mapping of families of sets, Soviet Math. Dokl. 11(1970), 116–120. On the Aleksandrov-Rassias problem for isometric mappings On the Aleksandrov-Rassias problem and the geometric invariance in Hilbert spaces S.-M. Jung and K.-S. Lee, An inequality for distances between 2n points and the Aleksandrov–Rassias problem, J. Math. Anal. Appl. 324(2)(2006), 1363–1369. S. Xiang, Mappings of conservative distances and the Mazur–Ulam theorem, J. Math. Anal. Appl. 254(1)(2001), 262–274. S. Xiang, On the Aleksandrov problem and Rassias problem for isometric mappings, Nonlinear Functional Analysis and Appls. 6(2001), 69-77. S. Xiang, On approximate isometries, in : Mathematics in the 21st Century (eds. K. K. Dewan and M. Mustafa), Deep Publs. Ltd., New Delhi, 2004, pp. 198–210.
Wikipedia:Aleksei Markushevich#0	Aleksei Ivanovich Markushevich (Russian: Алексей Иванович Маркушевич; 2 April [O.S. 20 March] 1908, Petrozavodsk – 7 June 1979, Moscow) was a Soviet mathematician, mathematical educator, and historian of mathematics. He is known for the Farrell–Markushevich theorem. == Biography == Markushevich's father worked as a junior architect for the Olonets provincial government. In 1916 the family moved to Semipalatinsk, where he graduated from secondary school in 1924. In 1925 he matriculated at the First Central Asian State University (now the National University of Uzbekistan). There in 1930 he graduated and was admitted to the correspondence course (education by mail) of the graduate school. In autumn 1931 he became a graduate student in the Research Institute of Mechanics and Mathematics, Moscow State University. There he received in 1934 his Aspirantur (similar to a PhD) under the supervision of Mikhail Lavrentyev, became at the beginning of 1935 a senior researcher, and became in 1938 an associate professor. At Moscow State University, Markushevich received in 1944 his Russian Doctor of Sciences degree (habilitation) and in 1946 became a full professor, retaining that academic position until his death. Markushevich was elected in 1945 a corresponding member and in 1950 a full member of Академии педагогических наук РСФСР (Academy of Pedagogical Sciences, Russian Soviet Federative Socialist Republic). For that academy he served as vice-president from 1950 to 1958 and again from 1964 to 1967. From 1958 to 1964 he was the primary deputy minister of education of the Russian Soviet Federative Socialist Republic. In Академии педагогических наук СССР (Academy of Pedagogical Sciences of the USSR), he was elected a full member in 1967 and served as its vice-president from 1967 to 1975. In 1978 he was an Invited Speaker at the International Congress of Mathematicians (ICM) in Helsinki. The last years of his life were overshadowed by a scandal related to the theft of medieval European manuscripts from Российский государственный архив древних актов, РГАДА / RGADA (Russian State Archives of Ancient Documents; investigations revealed that the main buyer of the stolen documents was Markushevich. He was buried in Moscow at the Kuntsevo Cemetery (section 10). == Mathematical research == Markushevich's mathematical research deals mainly with complex analysis, conformal mapping, and approximation theory. He published a series of papers on approximation, interpolation, and completeness; the papers contributed to the pioneering methods of functional analysis, in particular the theory of linear spaces, which began to be widely used in the theory of analytic functions. == Book publishing activities == He was head of the editorial office of mathematics of the Soviet Издательство технико-теоретической литературы (Publishing House of Technical and Theoretical Literature) from 1934 to 1937 and again from 1943 to 1947. He initiated and participated in the publication of the series of books Популярные лекции по математике (Popular Lectures on Mathematics) published in 62 volumes from 1950 to 1992 and the series of books Библиотека учителя (Teacher's Library). From 1951 to 1952 and from 1963 to 1966, he was among the initiators and editors of Энциклопедия элементарной математики (Encyclopedia of Elementary Mathematics). Markushevich was keenly interested in the history of books. Beginning in the 1940s he collected old books of historical interest. In 1976, he donated his collection of incunables to the State Library of the USSR named after V.I. Lenin. == Educational reform == During the 1960s and 1970s Markushevich was a leading supporter of reforming the teaching of mathematics in Soviet schools. In the 1960s, he took part in the creation of new school textbooks on mathematics, developed the pedagogical theory of school textbooks, and worked on improving the training of school mathematics teachers. He was the chair of the joint commission of the Academy of Sciences of the Soviet Union and the Academy of Pedagogical Sciences of the USSR, which determined the content of education in secondary schools. == Selected publications == === Articles === Die Arbeiten von C. F. Gauß über Funktionentheorie. (The works of C. F. Gauss on complex analysis) In: Hans Reichardt (ed.): C. F. Gauss. Gedenkband anlässlich des 100. Todestages am 23. Februar 1955. Teubner, Leipzig 1957, pp. 151–182. A. N. Kolmogorov, A. P. Yushkevich (eds.), B. L. Laptev, B. A. Rosenfel'd, A. I. Markushevich: Mathematics of the 19th Century. Geometry, analytic function theory. Birkhäuser, Basel 1996. Markushevich contributed the section Analytic Function Theory. === Books === Theory of Analytic Functions. Hindustan Publishing Co., Delhi, 1963. (translation of 1961 Russian edition, revised from the 1944 Russian edition) Skizzen zur Geschichte der analytischen Funktionen (Sketches on the history of analytic functions) (= Hochschulbücher für Mathematik. vol. 16). Deutscher Verlag der Wissenschaften, Berlin 1955. Theory of functions of a complex variable. 3 vols. Prentice-Hall, 1965–1967 Silverman, Richard A. (May 1984). Introductory Complex Analysis. Courier Corporation. ISBN 9780486646862. (1 volume abridgment of the 3 volumes translated by Richard A. Silverman) Entire Functions. American Elsevier, 1966, 105 pages (translated from the 1965 Russian edition); Markushevich, A. I. (12 May 2014). Entire Functions. Elsevier. ISBN 9781483274904. (ebook) The Remarkable Sine Functions. American Elsevier, 1966 (translated from 1965 Russian edition); Markushevich, A. I. (12 May 2014). The Remarkable Sine Functions. Elsevier. ISBN 9781483275215. (ebook) Markushevich, A. I. (1992). Introduction to the Classical Theory of Abelian Functions. American Mathematical Soc. ISBN 9780821841648. (ebook translated from the 1979 Russian edition by G. Bluher) == References == == External links == "Markushevich, Aleksei Ivanovich". mathnet.ru.
Wikipedia:Alena Varmužová#0	Alena Varmužová (24 April 1939 – 7 August 1997) was a Czech mathematician. She was specialized in creating teaching systems for mathematics education of young students. == Life and work == Alena Varmužová was born on 24 April 1939 in Rožnov pod Radhoštěm. She graduated from the Faculty of Science in Olomouc, having completed the coursework for mathematics and descriptive geometry. Then she enrolled in the Pedagogical Institute in Ostrava where she earned the title Doctor of Natural Sciences (1975), in 1988 she defended her candidate's thesis Content definition of the didactic system of mathematical preparation of preschool children, which dealt with designing systems to effectively teach mathematics to young students. In 1990, she was appointed associate professor for mathematics didactics at the Faculty of Education of the University of Ostrava, and she became head of the Department of Mathematics Didactics. She worked at the Union of Czech mathematicians and physicists. Varmužová authored the section of a textbook for secondary schools that was titled Reasoning Education - Mathematical Concepts. In 1991 she published her book, Mathematics for Preschool Children. Varmužová died on 7 August 1997 at the age of 58. Her husband was the sculptor Vratislav Varmuža. == References ==
Wikipedia:Alena Šolcová#0	Alena Šolcová (born 26 March 1950) is a Czech mathematician and science historian. She is the founder of the Kepler Museum, an astronomy museum in Prague. == Life and work == Between 1968 and 1973, Šolcová studied mathematics at the Faculty of Mathematics and Physics and Philosophy at Charles University. Between 2002 and 2005, she completed her doctoral studies in mathematics in Civil Engineering with the doctoral thesis titled Fermat's Ideas Revived in Mathematics Applied in Engineering, and in 2009 she completed her habilitation at the Czech Technical University in Prague, and was appointed associate professor in the field of applied mathematics. Šolcová works at the Faculty of Information Technologies of the Czech Technical University in Prague, where she teaches mathematical logic and the history of mathematics and computer science. She also deals with logic, number theory, some numerical methods and the history of mathematics, computer science and astronomy. Since 1992, she has led the SEDMA seminar (Seminar for the History of Mathematics, Informatics and Astronomy) and the working group for the history of exact sciences HEXA. She was the initiator and founder of an astronomy museum, the Kepler Museum in Prague (after its establishment, the museum was managed by the Czech Astronomical Society, in operation from 2009 to 2017). She initiated the establishment of commemorative plaques to Albert Einstein on Prague's Old Town Square and another to Austrian mathematician Johann Radon in Děčín, Czech Republic. Šolcová is an honorary member of the Union of Czech Mathematicians and Physicists and has served as its chairman beginning in 2018. She is also a member of the editorial board of the Czechoslovak Journal for Physics. She remains an active member of the Czech Mathematical Society and the Czech Society for Cybernetics and Informatics, which provides expertise concerning issues in logic, probability and reasoning. === Honors === The magic of numbers - From great discoveries to applications, Academia Praha, 2009, 2011, 2018 with co-authors Michal Křížek and Lawrence Somer, was awarded the Hlávká prize for scientific literature in 2010 (First prize). In 1998, the International Astronomical Union named planet No. 58682 Alenašolcová in her honor. == Selected publications == Šolcová, Alena, and Michał Křižek. "Fermat And Mersenne Numbers In Pepin's Test1." Demonstratio Mathematica 39, no. 4 (2006): 737–742. Křížek, Michal, Alena Šolcová, and Lawrence Somer. "Construction of Šindel sequences." Commentationes Mathematicae Universitatis Carolinae 48, no. 3 (2007): 373–388. Solcova, Alena, and Michal Krizek. "Vladimír Vand (1911–1968): Pioneer of Computational Methods in Crystallography." IEEE Annals of the History of Computing 33, no. 4 (2011): 38–44. Křížek, Michal, Lawrence Somer, and Alena Šolcová. From great discoveries in number theory to applications. Cham: Springer, 2021. == References == == External links == Official website
Wikipedia:Alessandra Carbone#0	Alessandra Carbone is an Italian mathematician and computer scientist. She is a professor in the computer science department of the Pierre and Marie Curie University. Since 2009 she has headed the laboratory of computational and quantitative biology. == Career == She gained her PhD in mathematics in 1993 at the Graduate Center of the City University of New York, supervised by Rohit Jivanlal Parikh, after which she took up a post doctoral post at the Paris Diderot University until 1995 when she took a position at the Technical University of Vienna until 1996. She has taught computer science at the Paris 12 Val de Marne University and the Institut des Hautes Études Scientifiques. She is currently a professor at the Pierre and Marie Curie University == Recognition == Carbone is a senior member of the Institut Universitaire de France and received the Irène Joliot-Curie Prize in 2010. She is also a recipient in 2012 of the Grammaticakis-Neumann Prize and the Legion of Honour in 2014. == References ==
Wikipedia:Alessandra Iozzi#0	Alessandra Iozzi (born 25 January 1959) is an Italian-born mathematician known for her research in geometric group theory. Originally from Rome, she holds Italian, Swiss, and American citizenships, and works as an adjunct professor of mathematics at ETH Zurich. == Education and career == Iozzi obtained a laurea at the Sapienza University of Rome in 1982, supervised by Massimo Picardello. Then, she moved to the University of Chicago where she earned a master's degree in 1985 and a Ph.D. in 1989. Her dissertation, Invariant Geometric Structures: A Non-Linear Extension of the Borel Density Theorem, was supervised by Robert Zimmer. After holding a lecturer position at the University of Pennsylvania for two years, she became a postdoctoral scholar first at the Mathematical Sciences Research Institute in Berkeley, CA and then at the Institute for Advanced Study in Princeton, NJ. From 1992 to 2000, she held a faculty position at the University of Maryland, College Park. She first came to ETH Zurich as a visiting researcher in 2000–2001. After holding professorships at the University of Strasbourg in France and the University of Basel in Switzerland, she returned to ETH Zurich as a senior scientist in 2006 and took her present faculty position there in 2008. == Recognition == Iozzi was named a Fellow of the American Mathematical Society, in the 2022 class of fellows, "for contributions to geometric group theory and the geometry of discrete subgroups of Lie groups, in particular higher Teichmuller theory." == References == == External links == Home page Alessandra Iozzi publications indexed by Google Scholar
Wikipedia:Alessandra Lunardi#0	Alessandra Lunardi (born 1958) is an Italian mathematician specializing in mathematical analysis. She is a professor in the department of mathematics and computer science at the University of Parma. She is particularly interested in Kolmogorov equations and free boundary problems. == Education and career == Lunardi was educated at the University of Pisa, completing her undergraduate studies there in 1980 and earning a Ph.D. there in 1983. Her dissertation, Analyticity of the maximal solution to fully nonlinear equations in Banach spaces, was supervised by Giuseppe Da Prato. After continuing on at Pisa as a researcher from 1984 to 1987, she was hired as a full professor at the University of Cagliari in 1987, and moved to Parma in 1994. == Contributions == Lunardi is the author of Analytic semigroups and optimal regularity in parabolic problems (Birkhäuser, 1995, reprinted 2013) and of Interpolation theory (Edizioni della Normale, 1998, 3rd ed., 2018). With G. Da Prato, P. C. Kunstmann, I. Lasiecka, R. Schnaubelt, and L. Weis, she is a co-author of Functional Analytic Methods for Evolution Equations (Springer, 2004). Lunardi is one of six editors-in-chief of the journal Nonlinear Differential Equations and Applications (NoDEA). She also served as editor-in-chief of Rivista di Matematica della Università di Parma for Series 7 of the journal, from 2002 to 2008. == Recognition == In 1987, Lunardi won the Bartolozzi Prize of the Italian Mathematical Union. In 2017, she won the Lucio & Wanda Amerio Gold Medal Prize of the Istituto Lombardo Accademia di Scienze e Lettere. == References == == External links == Home page
Wikipedia:Alessandra Sarti#0	Alessandra Sarti (born 1974) is an Italian mathematician specializing in algebraic geometry. She is the namesake of the Sarti surface, and has also published research on K3 surfaces. She works in France as a professor at the University of Poitiers and deputy director of the Institut national des sciences mathématiques et de leurs interactions (Insmi) of the French National Centre for Scientific Research in Paris. == Education and career == Sarti was born in 1974, in Ferrara, Italy. After studying for a laurea at the University of Ferrara from 1993 to 1997, she moved to Germany for graduate study in mathematics. After a year at the University of Göttingen, supported by an Italian research grant, she became a research assistant at the University of Erlangen–Nuremberg. She completed her Ph.D. there in 2001, with the dissertation Pencils of symmetric surfaces in P 3 {\displaystyle \mathbb {P} _{3}} , supervised by Wolf Barth. She took an assistant professor position at the University of Mainz in Germany, from 2003 to 2008, earning a habilitation there in 2007. After a temporary faculty position at the University of Erlangen–Nuremberg, she became a full professor at the University of Poitiers in France in 2008. At the University of Poitiers, she directed the Laboratoire de Mathématiques et Applications from 2016 to 2021. Since 2022, she has held a second affiliation as deputy director of the Institut national des sciences mathématiques et de leurs interactions of the French National Centre for Scientific Research (CNRS), in Paris. == Research == Sarti is the namesake of the Sarti surfaces (also called Sarti dodecics) a family of degree-12 nodal surfaces with 600 nodes that she discovered in 1999 and published in 2001.[SS] One member of the family can be chosen so that 560 of the nodes have real rather than complex coordinates. The Sarti surface has a K3 surface as one of its quotients, and some of Sarti's other publications include research on the symmetries of K3 surfaces.[K3a][K3b] === Selected publications === == Personal life == Sarti has a twin sister, Cristina Sarti, who also did a Ph.D. in mathematics in Germany. == References == == External links == Home page at the University of Poitiers
Wikipedia:Alessandro Padoa#0	Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive notion is truly independent of the other primitive notions. There is an analogous problem in axiomatic theories, namely deciding whether a given axiom is independent of the other axioms. The following description of Padoa's career is included in a biography of Peano: He attended secondary school in Venice, engineering school in Padua, and the University of Turin, from which he received a degree in mathematics in 1895. Although he was never a student of Peano, he was an ardent disciple and, from 1896 on, a collaborator and friend. He taught in secondary schools in Pinerolo, Rome, Cagliari, and (from 1909) at the Technical Institute in Genoa. He also held positions at the Normal School in Aquila and the Naval School in Genoa, and, beginning in 1898, he gave a series of lectures at the Universities of Brussels, Pavia, Berne, Padua, Cagliari, and Geneva. He gave papers at congresses of philosophy and mathematics in Paris, Cambridge, Livorno, Parma, Padua, and Bologna. In 1934 he was awarded the ministerial prize in mathematics by the Accademia dei Lincei (Rome). The congresses in Paris in 1900 were particularly notable. Padoa's addresses at these congresses have been well remembered for their clear and unconfused exposition of the modern axiomatic method in mathematics. In fact, he is said to be "the first … to get all the ideas concerning defined and undefined concepts completely straight". == Congressional addresses == === Philosophers' congress === At the International Congress of Philosophy Padoa spoke on "Logical Introduction to Any Deductive Theory". He says during the period of elaboration of any deductive theory we choose the ideas to be represented by the undefined symbols and the facts to be stated by the unproved propositions; but, when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions (instead of stating facts, that is, relations between the ideas represented by the undefined symbols) are simply conditions imposed upon undefined symbols. Then, the system of ideas that we have initially chosen is simply one interpretation of the system of undefined symbols; but from the deductive point of view this interpretation can be ignored by the reader, who is free to replace it in his mind by another interpretation that satisfies the conditions stated by the unproved propositions. And since the propositions, from the deductive point of view, do not state facts, but conditions, we cannot consider them genuine postulates. Padoa went on to say: ...what is necessary to the logical development of a deductive theory is not the empirical knowledge of the properties of things, but the formal knowledge of relations between symbols. === Mathematicians' congress === Padoa spoke at the 1900 International Congress of Mathematicians with his title "A New System of Definitions for Euclidean Geometry". At the outset he discusses the various selections of primitive notions in geometry at the time: The meaning of any of the symbols that one encounters in geometry must be presupposed, just as one presupposes that of the symbols which appear in pure logic. As there is an arbitrariness in the choice of the undefined symbols, it is necessary to describe the chosen system. We cite only three geometers who are concerned with this question and who have successively reduced the number of undefined symbols, and through them (as well as through symbols that appear in pure logic) it is possible to define all the other symbols. First, Moritz Pasch was able to define all the other symbols through the following four: 1. point 2. segment (of a line) 3. plane 4. is superimposable upon Then, Giuseppe Peano was able in 1889 to define plane through point and segment. In 1894 he replaced is superimposable upon with motion in the system of undefined symbols, thus reducing the system to symbols: 1. point 2. segment 3. motion Finally, in 1899 Mario Pieri was able to define segment through point and motion. Consequently, all the symbols that one encounters in Euclidean geometry can be defined in terms of only two of them, namely 1. point 2. motion Padoa completed his address by suggesting and demonstrating his own development of geometric concepts. In particular, he showed how he and Pieri define a line in terms of collinear points. == References == == Bibliography == A. Padoa (1900) "Logical introduction to any deductive theory" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 118–23. A. Padoa (1900) "Un Nouveau Système de Définitions pour la Géométrie Euclidienne", Proceedings of the International Congress of Mathematicians, tome 2, pages 353–63. Secondary: Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870–1940. Princeton Uni. Press. H.C. Kennedy (1980) Peano, Life and Works of Giuseppe Peano, D. Reidel ISBN 90-277-1067-8 . Suppes, Patrick (1957, 1999) Introduction to Logic, Dover. Discusses "Padoa's method." Smith, James T. (2000), Methods of Geometry, John Wiley & Sons, ISBN 0-471-25183-6 Jean Van Heijenoort (ed.) (1967) From Frege to Gödel. Cambridge: Harvard University Press == External links == O'Connor, John J.; Robertson, Edmund F., "Alessandro Padoa", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Alessio Corti#0	Alessio Corti (born 1965) is a Professor of Mathematics at Imperial College London working in Algebraic Geometry. Corti studied at the University of Pisa and Scuola Normale Superiore in Pisa, where he gained a diploma (Laurea) in 1987. He obtained his PhD in 1992 at the University of Utah under the supervision of János Kollár. As a post-doctoral researcher, he was at the Scuola Normale Superiore in Pisa and at the Mathematical Sciences Research Institute in Berkeley, California. From 1993 to 1996 he was the Dickson Instructor at the University of Chicago and in 1996 became lecturer, later reader, of mathematics at the University of Cambridge. Since 2005 he has been a professor at Imperial College London. In 2002, he was awarded the London Mathematical Society's Whitehead Prize. He is the originator of a project to create a periodic table of shapes. He is married and has a daughter, Beatrice (born 17 September 1992). == References == == External links == Imperial website Alessio Corti at the Mathematics Genealogy Project
Wikipedia:Alex James (mathematician)#0	Alex James is a British and New Zealand applied mathematician and mathematical biologist whose research involves the mathematical modelling of wildlife behaviour, gender disparities in academia, and the epidemiology of COVID-19. She is a professor in the school of mathematics and statistics at the University of Canterbury in New Zealand. == Education and career == After studying mathematics at Newcastle University in England, James earned a master's degree at University College London, and completed a PhD at the University of Leeds, working there with John Brindley on combustion engineering and catalytic converters. She became a lecturer at Sheffield Hallam University in 2001, and moved to the University of Canterbury in 2004. == Recognition == James was named a Fellow of the New Zealand Mathematical Society (NZMS) in 2015, and won the 2018 NZMS Research Award. She was on the team that won the Prime Minister's Science prize in 2020 and won the University of Canterbury Research medal jointly in 2021. She was awarded the NZIAM EO Tuck medal in 2024. == References == == External links == Home page Alex James publications indexed by Google Scholar
Wikipedia:Alexander Abrosimov#0	Alexánder Víktorovich Abrósimov (Russian: Александр Викторович Абросимов; November 16, 1948 – June 20, 2011) was a Russian mathematician and teacher. == Life == Dr. Abrosimov was born in 1948 in the city of Kuibyshev (now Samara). In 1971, he graduated from the Department of Mechanics and Mathematics of State University of Gorky (now Lobachevsky State University of Nizhny Novgorod). Dr. Abrosimov undertook graduate studies in the Department of Mechanics and Mathematics of Lomonosov Moscow State University under the supervision of Professor Boris Shabat. In 1984 he defended his Ph.D. dissertation “Complex Differential Systems and the Tangential Cauchy–Riemann Equations”. Dr. Abrosimov was Associate Professor in the Theory of Functions Subdepartment in the Department of Mechanics and Mathematics and Invited Lecturer in the Advanced School of General and Applied Physics (the base department of the Institute of Applied Physics and the Institute for Physics of Microstructures of the Russian Academy of Sciences). == Work == Beginning from his first works in 1971–3, Dr. Abrosimov studied overdetermined systems of partial differential equations where he successfully applied an original approach developed. A bright mathematical work in 1988 was devoted to an explicit procedure enabling one to decide whether two given smooth real hypersurfaces are locally CR-diffeomorphic. Subsequently, Dr. Abrosimov applied his original technique to describe CR-automorphisms of real quadrics of higher codimension. In this direction, he obtained important results and elaborated methods nowadays well known to CR-geometry specialists. First, Dr. Abrosimov proved that holomorphic automorphisms of a quadric of codimension two are furnished by birational transformations of degree two. Secondly, he convincingly demonstrated the power of the machinery of differential algebra in CR-geometry. In particular, he proved that under mild conditions the stabilizer of a point in the group of automorphisms of a quadric in Cn is a linear group. Thirdly, he was amongst the first researchers to look into scrutinizing a class of CR-manifolds of codimension one. To date, the class has remained a focus of active attention and efforts of research. Overall, Dr. Abrosimov published more than 25 scientific works on complex analysis. Some of the works of Dr. Abrosimov in CR-geometry and adjacent fields in complex analysis are deemed pioneering and his contribution important. == Selected papers == 1. A.V. Abrosimov and L.G. Mikhailov (1971). On Some Overdetermined Systems in Partial Derivatives. Proceedings of the Academy of Sciences of the Tadjik SSR, Vol. IV, No. 6, (8 pages). 2. A.V. Abrosimov and L.G. Mikhailov (1973). Generalized Cauchy–Riemann System in Many Independent Complex Variables. Proceedings of the USSR Academy of Sciences, Vol. 210, No. 1 (4 pages). 3. A.V. Abrosimov (1977). Beltrami System in Many Independent Complex Variables. Proceedings of the USSR Academy of Sciences, Vol. 236, No. 6 (4 pages). 4. A.V. Abrosimov (1983). Complex Differential Systems and the Cauchy–Riemann Tangent Equations. Sbornik: Mathematics. Vol. 122, No. 4 (16 pages). 5. A.V. Abrosimov (1988). On Locally Biholomorphic Equivalence of Smooth Hypersurfaces in С2. Proceedings of the USSR Academy of Sciences, Vol. 299, No. 4 (5 pages). 6. A.V. Abrosimov (1995). A Description of Locally Biholomorphic Automorphisms of Standard Quadrics of Codimension Two. American Mathematical Society, 1064–5616/95 (42 pages). 7. A.V. Abrosimov (2003). Linearity of Standard Quadrics of Codimension m in Cn+m . Mathematical Notes, No. 1 (5 pages). == References == == External links == Russian Wikipedia article on Abrosimov Obituary in the Notices of the American Mathematical Society http://upload.wikimedia.org/wikipedia/commons/f/f3/Alexander_abrosimov_notices_of_AMS.pdf
Wikipedia:Alexander Anderson (mathematician)#0	Alexander Anderson (c. 1582 in Aberdeen – c. 1620 in Paris) was a Scottish mathematician. == Life == He was born in Aberdeen, possibly in 1582, according to a print which suggests he was aged 35 in 1617. It is unknown where he was educated, but it is likely that he initially studied writing and philosophy (the "belles lettres") in his home city of Aberdeen. He then went to the continent, and was a professor of mathematics in Paris by the start of the seventeenth century. There he published or edited, between the years 1612 and 1619, various geometric and algebraic tracts. He described himself as having "more wisdom than riches" in the dedication of Vindiciae Archimedis (1616). He was first cousin of David Anderson of Finshaugh, a celebrated mathematician, and David Anderson's daughter was the mother of mathematician James Gregory. == Work == He was selected by the executors of François Viète to revise and edit Viète's manuscript works. Viète died in 1603, and it is unclear if Anderson knew him, but his eminence was sufficient to attract the attention of the dead man's executors. Anderson corrected and expanded upon Viète's manuscripts, which extended known geometry to the new algebra, which used general symbols to represent quantities. === Publications === The known works of Anderson amount to six thin quarto volumes, and as the last of them was published in 1619, it is probable that the author died soon after that year, but the precise date is unknown. He wrote other works that have since been lost. From his last work it appears he wrote another piece, "A Treatise on the Mensuration of Solids," and copies of two other works, Ex. Math. and Stereometria Triangulorum Sphæricorum, were in the possession of Sir Alexander Hume until the after the middle of the seventeenth century. 1612: Supplementum Apollonii Redivivi 1615: Ad Angularum Sectionem Analytica Theoremata F. Vieta 1615: Pro Zetetico Apolloniani 1615: Francisci Vietae Fontenaeensis 1616: Vindiciae Archimedis 1619: Alexandri Andersoni Exercitationum Mathematicarum Decas Prima == See also == Marin Getaldić Denis Henrion Frans van Schooten == References == Attribution: This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Anderson, Alexander". Encyclopædia Britannica. Vol. 1 (11th ed.). Cambridge University Press. p. 959. == Further reading == Chambers, Robert; Thomson, Thomas Napier (1857). "Anderson, Alexander" . A Biographical Dictionary of Eminent Scotsmen. Vol. 1. Glasgow: Blackie and Son. p. 53 – via Wikisource.
Wikipedia:Alexander Barvinok#0	Alexander I. Barvinok (born March 27, 1963) is a Russian American mathematician and a professor of mathematics at the University of Michigan. Barvinok received his Ph.D. from St. Petersburg State University in 1988 under the supervision of Anatoly Moiseevich Vershik. In 1999, Barvinok received the Presidential Early Career Award for Scientists and Engineers (PECASE) from President Bill Clinton. Barvinok gave an invited talk at the 2006 International Congress of Mathematicians in Madrid. In 2012, Barvinok became a Fellow of the American Mathematical Society. In 2023, Barvinok left the American Mathematical Society by refusing to renew his membership in protest of its non-opposition to "DEI statements" and "compelled language", referencing his experiences in the Soviet Union. == References ==
Wikipedia:Alexander Braverman#0	Alexander Braverman (born June 8, 1974) is an Israeli mathematician. == Life and work == Braverman was born in Moscow.. He earned in 1993 a BA degree in mathematics from the University of Tel Aviv, where in 1998 he received a Ph.D. (Kazhdan-Laumon Representations of Finite Chevalley Groups, Character Sheaves and Some Generalization of the Lefschetz-Verdier Trace Formula) under supervision of Joseph Bernstein. From 1997 to 1999 he was a C.L.E. Moore instructor at Massachusetts Institute of Technology and in 2004 Benjamin Peirce Lecturer at Harvard University. He was an associate professor at Brown University from 2004 to 2009 and then a full professor from 2009 to 2015. He is a full professor at University of Toronto since 2015 and an associate faculty member at Perimeter Institute for Theoretical Physics. He was also a visiting scholar at Institute for Advanced Study (1997, 1999), the University of Paris VI and the Paris-Nord, the Hebrew University in Jerusalem, the Weizmann Institute, Clay Mathematics Institute and at the IHES in Paris. Braverman specializes in the geometric Langlands program, the intersection of number theory, algebraic geometry and representation theory, which also has applications to mathematical physics. In 2006 he was an invited speaker at the International Congress of Mathematicians in Madrid (Spaces of quasi-maps into the flag varieties and their applications). == References == == External links == CV at Brown University
Wikipedia:Alexander Brown (mathematician)#0	Alexander Brown FRSE FRSSAf (1878–1947) was a Scottish-born mathematician and educator in South Africa. He contributed to the study of the ratio of incommensurables in geometry and relations between the distances of a point from three vertices of a regular polygon. == Career == Brown was born in Dalkeith, near Edinburgh, Scotland on 5 May 1878. He attended Newton Public School and then George Watson's College, where he was the winner of the Wright Bursary in his final year. He matriculated at the University of Edinburgh in 1893 where he obtained a MA (Mathematics and Natural Science)(First Class) and a BSc (Mathematics and Natural Science)(Special Distinction) in 1897. While at Edinburgh University he held the first Heriot Bursary and Mackay Smith scholarship. On completion of his degrees he won the Vans Dunlop scholarship, the Baxter scholarship and the Drummond scholarship. He was mathematical master at the High School of Dundee from 1897 to 1899. In 1899 he won the Ferguson Scholarship in Mathematics and he went to Gonville and Caius College, Cambridge where he obtained a First Class in Mathematics in 1902. In 1903 he became Deputy Professor of Applied Mathematics and Physics for one year at the South African College, Cape Town in the absence of the holder of the chair, Carruthers Beattie, who was on a year-long magnetic survey of South Africa. In recognition of Brown's exceptional work during that year the College Council split the Chair and Beattie was offered the professorship of Physics while Brown was offered the professorship of Mathematics. He retained this position until his death on 27 January 1947. He was elected a Fellow of the Royal Society of Edinburgh in 1907. His proposers were George Chrystal, Arthur John Pressland, John Sturgeon Mackay and John Alison. In 1918 he was elected a Fellow of the Royal Society of South Africa. == Personal life == He married Mary Graham in 1911 and they had a daughter and a son. In his leisure time he was a great reader and a keen musician. == Works == He contributed to the study of the ratio of incommensurables in geometry and relations between the distances of a point from three vertices of a regular polygon. His paper on this subject was read before the Edinburgh Mathematical Society on 11 June 1909. == Selected publications == Brown, A. (1905) "Convergence of a Reversed Power Series" British Association Report Brown, A. (1907) "Examination of the Validity of an Approximate Solution of a Certain Velocity Equation" Transaction of the South African Philosophical Society, vol xvi, pt. 3 Brown, A. (1916) "The Equivalent Mass of a Spring Vibrating Longitudinally" Transactions of the Royal Society of South Africa vol. v, p. 565. Brown, A. (1916) "The Arrangement of Successive Convergents in the Order of Accuracy" Transactions of the Royal Society of South Africa vol. v, p. 653. Brown, A. (1916) "The Use of a Standard Parabola for Drawing Diagrams of Bending Moment and of Shear in a Beam Uniformly Loaded" Transactions of the Royal Society of South Africa vol. v, p. 659. == External links == Alexander Brown Obituary notice in Transactions of the Royal Society of South Africa Biography of Alexander Brown on MacTutor Plug, C. "Brown, Prof Alexander (applied mathematics)". S2A3 Biographical Database of Southern African Science. Retrieved 21 December 2019. == References ==
Wikipedia:Alexander Dranishnikov#0	Alexander Nikolaevich Dranishnikov (Александр Николаевич Дранишников, born 5 February 1958) is a Russian-American mathematician, focusing in geometry and topology, currently a Distinguished Professor at the University of Florida. In 1998 he was an Invited Speaker at the International Congress of Mathematicians in Berlin. In 2012 he became one of the inaugural fellows of the American Mathematical Society. == References == == External links == Dranishnikov Alexander Nikolaevich, mathnet.ru
Wikipedia:Alexander Dyakonov#0	Alexander Dyakonov (Russian: Алекса́ндр Генна́дьевич Дья́конов) (born 1979) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University. Professor of the Russian Academy of Sciences. Multiple winner of international competitions in applied data analysis. He defended the thesis «Algebraic closures of the generalized model of recognition algorithms based on the calculation of estimates» for the degree of Doctor of Physical and Mathematical Sciences (2010). Author of two books and more than 50 scientific articles. == References == == Bibliography == Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. 2010. pp. 358–359. ISBN 978-5-211-05838-5 – via Author-compiler Evgeny Grigoriev. == External links == Annals of the Moscow University(in Russian) Resolution of the Presidium of the RAS(in Russian) MSU CMC(in Russian) Scientific works of Alexander Dyakonov Scientific works of Alexander Dyakonov(in English)
Wikipedia:Alexander Furman#0	Alexander Furman (Hebrew: אלכסנדר פורמן) is a mathematician at the University of Illinois, Chicago. Furman received his bachelor's degree in mathematics and computer science from the Hebrew University of Jerusalem in 1986, later earning his master's degree and PhD in mathematics in 1989 and 1996, respectively, from the same university. == Career == Furman started teaching mathematics in 1996 as an L. E. Dickson Instructor of Mathematics at the University of Chicago. In 1997, he received a position as a postdoctoral fellow at Penn State University. He has worked at the University of Illinois Chicago since 1997, serving as an assistant professor from 1999 until 2007 and beginning as a full-time professor in 2007. Furman also runs the UIC Math Olympiad Project where he works with high school-age students, discussing and working out mathematical problems. == Honors and awards == Furman has been awarded multiple National Science Foundation and Binational Science Foundation grants. In 2014, Furman was made a Simons Foundation Fellow in Mathematics and was awarded the National Science Foundation CAREER award for his work in teaching through research. The same year, he was an invited speaker for the International Congress of Mathematics hosted in Seoul. For his work in dynamical systems, ergodic theory, and Lie groups, he was made an American Mathematical Society Fellow in 2016. He has additionally received UIC's University Scholar Award, as well as the LAS Distinguished Professor Award. == References ==
Wikipedia:Alexander Gammerman#0	Alexander Gammerman (born 2 November 1944) is a British computer scientist, and professor at Royal Holloway University of London. He is the co-inventor of conformal prediction. He is the founding director of the Centre for Machine Learning at Royal Holloway, University of London, and a Fellow of the Royal Statistical Society. == Career == Gammerman's academic career has been pursued in the Soviet Union and the United Kingdom. He started working as a Research Fellow in the Agrophysical Research Institute, St. Petersburg. In 1983, he emigrated to the United Kingdom and was appointed as a lecturer in the Computer Science Department at Heriot-Watt University, Edinburgh. Together with Roger Thatcher, Gammerman published several articles on Bayesian inference. In 1993, he was appointed to the established chair in Computer Science at University of London tenable at Royal Holloway and Bedford New College, where he served as the Head of Computer Science department from 1995 to 2005. In 1998, the Centre for Reliable Machine Learning was established, and Gammerman became the first director of the centre. Gammerman has published 7 books, more than 150 research papers, and has an estimated h-index of 34. He ranks amongst the top 1% researcher in artificial intelligence and machine learning. == Honours and awards == In 1996, Gammerman received the P.W. Allen Award from the Forensic Science Society. In 2006, he became a Honorary Professor, at University College London. In 2009, he became a Distinguished Professor at Complutense University of Madrid, Spain. In 2019, he received a research grant funded by the energy company Centrica about predicting the time to the next failure of equipment. In 2020, he received the Amazon Research Award for the project titled Conformal Martingales for Change-Point Detection == Selected books == Measures of Complexity (2016), Springer, ISBN 3319357786. Algorithmic Learning in a Random World (2005), Springer, ISBN 0387001522. Causal Models and Intelligent Data Management (1999), Springer, ISBN 978-3-642-58648-4. Probabilistic Reasoning and Bayesian Belief Networks (1998), Nelson Thornes Ltd, ISBN 1872474268. Computational Learning and Probabilistic Reasoning (1996), Wiley, ISBN 0471962791. == References == == External links == Gammerman's University Website
Wikipedia:Alexander Givental#0	Alexander Givental (Russian: Александр Борисович Гивенталь) is a Russian-American mathematician who is currently Professor of Mathematics at the University of California, Berkeley. His main contributions have been in symplectic topology and singularity theory, as well as their relation to topological string theories. Givental graduated from the famed Moscow high school #2 (Лицей «Вторая школа»), but was not able to enter a program at a top university due to antisemitism in Soviet mathematics. He completed his undergraduate and master studies at the Gubkin Russian State University of Oil and Gas, and defended his Ph.D. under the supervision of V. I. Arnold in 1987. He emigrated to the United States in 1990. He provided the first proof of the mirror conjecture for Calabi–Yau manifolds that are complete intersections in toric ambient spaces, in particular for quintic hypersurfaces in P4. As an extracurricular activity, he translates Russian poetry into English and publishes books, including his own translation of a textbook (Элементарная геометрия (Киселёв)) in geometry by Andrey Kiselyov and poetry of Marina Tsvetaeva. Givental is a father of two. == References == Cox, David A.; Katz, Sheldon (1999), Mirror Symmetry and Algebraic Geometry, Providence, Rhode Island: American Mathematical Society, ISBN 0-8218-1059-6. Sumizdat, publisher of English translation of Geometry MAA review of Geometry == External links == Personal website at Berkeley
Wikipedia:Alexander Goncharov#0	Alexander B. Goncharov (born April 7, 1960) is a Soviet American mathematician and the Philip Schuyler Beebe Professor of Mathematics at Yale University. He won the EMS Prize in 1992. Goncharov won a gold medal at the International Mathematical Olympiad in 1976. He attained his doctorate at Lomonosov Moscow State University in 1987, under supervision of Israel Gelfand with thesis Generalized conformal structures on manifolds. Goncharov was an Invited Speaker at the 1994 International Congress of Mathematicians and gave a talk Polylogarithms in arithmetic and geometry. In 2019, Goncharov was appointed the Philip Schuyler Beebe Professor of Mathematics at Yale University, as well as the Gretchen and Barry Mazur Chair at the Institut des hautes études scientifiques. == Selected publications == Goncharov, A.B. (1995). "Geometry of configurations, polylogarithms, and motivic cohomology". Advances in Mathematics. 114 (2): 197–318. doi:10.1006/aima.1995.1045. (with A. M. Levin) Goncharov, A. B.; Levin, A. M. (1998). "Zagier's conjecture on L(E,2)". Inventiones Mathematicae. 132 (2): 393–432. Bibcode:1998InMat.132..393G. doi:10.1007/s002220050228. Goncharov, Alexander (1999). "Volumes of hyperbolic manifolds and mixed Tate motives". Journal of the American Mathematical Society. 12 (2): 569–618. doi:10.1090/S0894-0347-99-00293-3. (with P. Deligne) Deligne, P.; Goncharov, A. (2005), "Groupes fondamentaux motiviques de Tate mixte", Annales Scientifiques de l'École Normale Supérieure, 38 (1): 1–56, arXiv:math/0302267, doi:10.1016/j.ansens.2004.11.001, S2CID 119144154 (with V. V. Fock) Fock, Vladimir; Goncharov, Alexander (2006). "Moduli spaces of local systems and higher Teichmüller theory". Publications Mathématiques de l'IHÉS. 103: 1–211. arXiv:math/0311149. doi:10.1007/s10240-006-0039-4. (with V. V. Fock) Fock, V.V.; Goncharov, A.B. (2009). "The quantum dilogarithm and representations of quantum cluster varieties". Inventiones Mathematicae. 175 (2): 223–286. arXiv:math/0702397. doi:10.1007/s00222-008-0149-3. (with H. Gangl, A. Levin) Gangl, Herbert; Goncharov, Alexander B.; Levin, Andrey (2007). "Multiple logarithms, algebraic cycles and trees". In Cartier, Pierre; Moussa, Pierre; Julia, Bernard; Vanhove, Pierre (eds.). Frontiers in Number Theory, Physics and Geometry. Vol. 2. Springer Verlag. pp. 759–774. arXiv:math/0504552. doi:10.1007/978-3-540-30308-4_16. ISBN 978-3-540-30307-7. S2CID 201209. (with V. V. Fock) Goncharov, Alexander B.; Fock, Vladimir V. (2009). "Cluster ensembles, quantization and the dilogarithm". Annales Scientifiques de l'École Normale Supérieure. 42 (6): 865–930. arXiv:math/0311245. doi:10.24033/asens.2112. (with R. Kenyon) Kenyon, Richard; Goncharov, Alexander B. (2013). "Dimers and cluster integrable systems". Annales Scientifiques de l'École Normale Supérieure. 46 (5): 747–813. arXiv:1107.5588. doi:10.24033/asens.2201. S2CID 17940641. (with T. Dimofte, M. Gabella) Dimofte, Tudor; Gabella, Maxime; Goncharov, Alexander B. (2016). "K-decompositions and 3d gauge theories". Journal of High Energy Physics. 2016 (11): #151. arXiv:1301.0192. Bibcode:2016JHEP...11..151D. doi:10.1007/JHEP11(2016)151. (with J. Golden, M. Spradlin, C. Vergu, A. Volovich) Golden, J. K.; Goncharov, A. B.; Spradlin, M.; Vergu, C.; Volovich, A. (2014). "Motivic Amplitudes and Cluster Coordinates". Journal of High Energy Physics. 2014: #91. arXiv:1305.1617. Bibcode:2014JHEP...01..091G. doi:10.1007/JHEP01(2014)091. == See also == Goncharov conjecture == References == == External links == Website at Yale University Alexander Goncharov at the Mathematics Genealogy Project Alexander Goncharov's results at International Mathematical Olympiad
Wikipedia:Alexander Gorban#0	Alexander Nikolaevich Gorban (Russian: Александр Николаевич Горба́нь) is a scientist of Russian origin, working in the United Kingdom. He is a professor at the University of Leicester, and director of its Mathematical Modeling Centre. Gorban has contributed to many areas of fundamental and applied science, including statistical physics, non-equilibrium thermodynamics, machine learning and mathematical biology. Gorban is the author of about 20 books and 300 scientific publications. He has founded several scientific schools in the areas of physical and chemical kinetics, dynamical systems theory and artificial neural networks, and is ranked as one of the 1000 most cited researchers of Russian origin. In 2020, Gorban presented a keynote talk at the IEEE World Congress on Computational Intelligence. Gorban has supervised 6 habilitations and more than 30 PhD theses. == Biography == Gorban was born in Omsk, on 19 April 1952. His father Nikolai Vasilievich Gorban was a historian and writer exiled to Siberia, and his mother was a literature teacher in Omsk Pedagogical Institute. From 1965 to 1966, he studied at the Specialized Educational Scientific Center on Physics, Mathematics, Chemistry and Biology of Novosibirsk State University (SESC NSU). In 1967, at the age of 15, he entered Novosibirsk State University but was excluded from it in autumn 1969 because of his participation in January 1968 in political student movements against the convictions of Soviet writers Alexander Ginzburg and Yuri Galanskov. After studying for a year in a vocational technical school and following an individual extramural program at Omsk Pedagogical Institute, he obtained a master's degree with a thesis entitled Sets of removable singularities in Banach spaces and continuous maps under the supervision of Russian mathematician Vladimir B. Melamed. From 1973 to 1976, he worked in the Omsk Institute Of Transport Engineers and published his first scientific works, but his scientific career could not develop successfully because of his past political record. He had several temporary work places from 1976 to 1978, each time being compelled to resign, but then moved to Krasnoyarsk where he was permanently employed at the Institute of Computational Modeling. In 1980, Gorban obtained his Candidate of Sciences diploma, corresponding to PhD in the Russian scientific degree hierarchy. His thesis title was Slow relaxations and bifurcations of omega-limit sets of dynamical systems (translated later into English ). His viva was organized by Olga Ladyzhenskaya, Mark Krasnosel'skii, and George M. Zaslavsky. With the beginning of Perestroika he became the head of the Laboratory of Non-Equilibrium Systems in 1989 and completed his habilitation in 1990. In 1995 he became the deputy director of the Institute of Computational Modeling and head of the Computational Mathematics Department. At the same time, he taught at Krasnoyarsk State University (1981-1991) and subsequently headed the Neuroinformatics Department at the Krasnoyarsk State Technical University (1993-2006). In the 1990s, Gorban visited several mathematical institutes in US and Europe, including the Clay Mathematics Institute, Courant Institute of Mathematical Sciences, Institut des Hautes Etudes Scientifiques, ETH (2003-2004), Isaac Newton Institute. In 2004, Gorban became Professor of Applied Mathematics at the Leicester University, UK, and the chair of its Mathematical Modeling Centre. Gorban is a stepbrother of Svetlana Kirdina. == Research activity == Gorban's scientific contributions have been made in theoretical physics, mechanics, functional analysis, theory of natural selection, theory of adaptation, artificial neural networks, physical kinetics, bioinformatics. A top level view of scientific activity and the future of applied mathematics have been given in his book "Demon of Darwin: idea of optimality and natural selection", articles and public lectures. In functional analysis, Gorban has investigated the properties of analytical Fredholm subsets in Banach spaces, formulated the relevant principle of maximum modulus and proved an analogue of the Remmert-Stein theorem. In mathematical chemistry, Gorban has investigated the thermodynamical properties of chemical systems based on the analysis of Lyapunov's function trees in the polytope of conservation laws. He developed a theory of thermodynamically admissible paths for complex multidimensional systems of chemical thermodynamics and kinetics. Together with Grigoriy Yablonsky and his team he developed methods of mathematical modeling and analysis of chemical system models for kinetics of catalytic reactions. He investigated the relaxation properties of some chemical systems and developed the singularity theory for transient processes of dynamical systems, developed the method of path summation for solving the chemical kinetics equations, developed a theory of dynamic limitation and asymptotology of chemical reaction networks which was applied to modeling of biological signalling networks and mechanisms of microRNA action on translation regulation. Gorban has developed a series of methods for solving equations of chemical and physical kinetics, based on constructive methods of invariant manifold approximation. This theory has found many applications in the construction of physically consistent hydrodynamics as a part of Hilbert's sixth problem, modeling non-equilibrium flows, in the kinetic theory of phonons, for model reduction in chemical kinetics, and modeling liquid polymers. He developed new methods for application of the Lattice Boltzmann's Method, based on its thermodynamical properties. Gorban has developed a mathematical model of the Gorlov helical turbine and estimated its achievable efficiency in energy capture. He investigated general problems of geometrical interpretation of thermodynamics and general properties of non-classical entropies. In the mathematical theory of natural selection, Gorban developed a theory of a special class of dynamical systems with inheritance. He discovered and explained theoretically the universal phenomenon of system adaptation under stress conditions, leading to simultaneous increase of correlations and variance in the multidimensional space of system parameters. The Anna Karenina principle developed by Gorban is now applied as a method of diagnostics and prognosis for economics and human physiology. Gorban developed highly efficient parallel methods for artificial neural networks (ANN) learning, based on systematic use of duality of their functioning, and developed methods of knowledge extraction from data based on sparse ANNs. He proved the theorem of universal approximation properties of ANN. All these approaches have found numerous applications in existing expert systems. Together with I. Tyukin, he developed a series of methods and algorithms for fast, non-iterative and non-destructive corrections of errors in legacy Artificial intelligence systems. These methods are based on the concentration of measure phenomena, ideas of statistical mechanics and original stochastic separation theorems. In applied statistics, Gorban developed methods for constructing principal manifolds (Elastic maps method) and their generalizations (principal graphs, principal trees), based on the mechanical analogy with elastic membrane. The method has found numerous applications for visualization and analysis of economical, sociological and biological data. In collaboration with E.M. Mirkes and other authors, used machine learning methods for connecting individual psychological characteristics and predisposition to consuming certain types of drugs. In bioinformatics Gorban was one of the first to apply the method of frequency dictionaries and Principle of maximum entropy for analysis of nucleotide and amino acid sequences. He investigated the general properties of compact genomes and proved the existence of a 7-cluster structure in the genome sequence, which was applied for solving the de novo gene identification problem. == Bibliography == Selected books: Selected articles: == Notes == == External links == Alexander N. Gorban's homepage Correlation adaptometry Thermodynamics in chemical reaction networks Methods of non-linear principal manifolds The mathematical theory of natural selection
Wikipedia:Alexander Grothendieck#0	Alexander Grothendieck, later Alexandre Grothendieck in French (; German: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] ; French: [ɡʁɔtɛndik]; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry, homological algebra, and K-theory. He later became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political and religious pursuits (first Buddhism and later, a more Catholic Christian vision). In 1991, he moved to the French village of Lasserre in the Pyrenees, where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014. == Life == === Family and childhood === Grothendieck was born in Berlin to anarchist parents. His father, Alexander "Sascha" Schapiro (also known as Alexander Tanaroff), had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant German family in Hamburg and worked as a journalist. As teenagers, both of his parents had broken away from their early backgrounds. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and initially, his birth name was recorded as "Alexander Raddatz." That marriage was dissolved in 1929 and Schapiro acknowledged his paternity, but never married Hanka Grothendieck. Grothendieck had a maternal sibling, his half sister Maidi. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism. His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg. According to Winfried Scharlau, during this time, his parents took part in the Spanish Civil War as non-combatant auxiliaries. However, others state that Schapiro fought in the anarchist militia. === World War II === In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward his father was interned in Le Vernet. He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners." The first camp was the Rieucros Camp, where his mother may have contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend the local school, at Mende. Once, he managed to escape from the camp, intending to assassinate Hitler. Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Grothendieck was permitted to live separated from his mother. In the village of Le Chambon-sur-Lignon, he was sheltered and hidden in local boarding houses or pensions, although he occasionally had to seek refuge in the woods during Nazi raids, surviving at times without food or water for several days. His father was arrested under the Vichy anti-Jewish legislation, and sent to the Drancy internment camp, and then handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Le Chambon, Grothendieck attended the Collège Cévenol (now known as the Le Collège-Lycée Cévenol International), a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Le Chambon attended Collège Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics. In 1990, for risking their lives to rescue Jews, the entire village was recognized as "Righteous Among the Nations". === Studies and contact with research mathematics === After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where at first he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948. Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, topological vector spaces: Jean Dieudonné and Laurent Schwartz. The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paper La dualité dans les espaces (F) et (LF); it ended with a list of 14 open questions, relevant for locally convex spaces. Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months. In Nancy, he wrote his dissertation under those two professors on functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. In 1953 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 to March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it is there that he completed his last major work on that topic (on "metric" theory of Banach spaces). Grothendieck moved to Lawrence, Kansas at the beginning of 1955, and there he set his old subject aside in order to work in algebraic topology and homological algebra, and increasingly in algebraic geometry. It was in Lawrence that Grothendieck developed his theory of abelian categories and the reformulation of sheaf cohomology based on them, leading to the very influential "Tôhoku paper". In 1957 he was invited to visit Harvard University by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government—a refusal which, he was warned, threatened to land him in prison. The prospect of prison did not worry him, so long as he could have access to books. Comparing Grothendieck during his Nancy years to the École Normale Supérieure-trained students at that time (Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, and Bernard Malgrange), Leila Schneps said: He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics. His first works on topological vector spaces in 1953 have been successfully applied to physics and computer science, culminating in a relation between Grothendieck inequality and the Einstein–Podolsky–Rosen paradox in quantum physics. === IHÉS years === In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. Grothendieck attracted attention by an intense and highly productive activity of seminars there (de facto working groups drafting into foundational work some of the ablest French and other mathematicians of the younger generation). Grothendieck practically ceased publication of papers through the conventional, learned journal route. However, he was able to play a dominant role in mathematics for approximately a decade, gathering a strong school. Officially during this time, he had as students Michel Demazure (who worked on SGA3, on group schemes), Monique Hakim (relative schemes and classifying topos), Luc Illusie (cotangent complex), Michel Raynaud, Michèle Raynaud, Jean-Louis Verdier (co-founder of the derived category theory), and Pierre Deligne. Collaborators on the SGA projects also included Michael Artin (étale cohomology), Nick Katz (monodromy theory, and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of nonabelian cohomology there as well. Many others such as David Mumford, Robin Hartshorne, Barry Mazur and C.P. Ramanujam were also involved. === "Golden Age" === Alexander Grothendieck's work during what is described as the "Golden Age" period at the IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory, and complex analysis. His first (pre-IHÉS) discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced K-theory. Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique (EGA) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it. Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology (relevant also in categorical logic). He also provided, by means of a categorical Galois theory, an algebraic definition of fundamental groups of schemes giving birth to the now famous étale fundamental group and he then conjectured the existence of a further generalization of it, which is now known as the fundamental group scheme. As a framework for his coherent duality theory, he also introduced derived categories, which were further developed by Verdier. The results of his work on these and other topics were published in the EGA and in less polished form in the notes of the Séminaire de géométrie algébrique (SGA) that he directed at the IHÉS. === Political activism === Grothendieck's political views were radical and pacifistic. He strongly opposed both United States intervention in Vietnam and Soviet military expansionism. To protest against the Vietnam War, he gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed. In 1966, he had declined to attend the International Congress of Mathematicians (ICM) in Moscow, where he was to receive the Fields Medal. He retired from scientific life around 1970 after he had found out that IHÉS was partly funded by the military. He returned to academia a few years later as a professor at the University of Montpellier. While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from the IHÉS, those who knew him say that the causes of the rupture ran more deeply. Pierre Cartier, a visiteur de longue durée ("long-term guest") at the IHÉS, wrote a piece about Grothendieck for a special volume published on the occasion of the IHÉS's fortieth anniversary. In that publication, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and the downtrodden. As Cartier puts it, Grothendieck came to find Bures-sur-Yvette as "une cage dorée" ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a mandarin of the scientific world. In addition, after several years at the IHÉS, Grothendieck seemed to cast about for new intellectual interests. By the late 1960s, he had started to become interested in scientific areas outside mathematics. David Ruelle, a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics. Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics. In 1970, Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel, created a political group entitled Survivre—the name later changed to Survivre et vivre. The group published a bulletin and was dedicated to antimilitary and ecological issues. It also developed strong criticism of the indiscriminate use of science and technology. Grothendieck devoted the next three years to this group and served as the main editor of its bulletin. Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left the IHÉS. After leaving the IHÉS, Grothendieck became a temporary professor at Collège de France for two years. He then became a professor at the University of Montpellier, where he became increasingly estranged from the mathematical community. He formally retired in 1988, a few years after having accepted a research position at the CNRS. === Manuscripts written in the 1980s === While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content. Produced during 1980 and 1981, La Longue Marche à travers la théorie de Galois (The Long March Through Galois Theory) is a 1600-page handwritten manuscript containing many of the ideas that led to the Esquisse d'un programme. It also includes a study of Teichmüller theory. In 1983, stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University, Grothendieck wrote a 600-page manuscript entitled Pursuing Stacks. It began with a letter addressed to Daniel Quillen. This letter and successive parts were distributed from Bangor (see External links below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks. The manuscript, which is being edited for publication by G. Maltsiniotis, later led to another of his monumental works, Les Dérivateurs. Written in 1991, this latter opus of approximately 2000 pages, further developed the homotopical ideas begun in Pursuing Stacks. Much of this work anticipated the subsequent development during the mid-1990s of the motivic homotopy theory of Fabien Morel and Vladimir Voevodsky. In 1984, Grothendieck wrote the proposal Esquisse d'un Programme ("Sketch of a Programme") for a position at the Centre National de la Recherche Scientifique (CNRS). It describes new ideas for studying the moduli space of complex curves. Although Grothendieck never published his work in this area, the proposal inspired other mathematicians to work in the area by becoming the source of dessin d'enfant theory and anabelian geometry. Later, it was published in two-volumes and entitled Geometric Galois Actions (Cambridge University Press, 1997). During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA on Bertini-type theorems (EGA V, published in Ulam Quarterly in 1992–1993 and later made available on the Grothendieck Circle web site in 2004). In the extensive autobiographical work, Récoltes et Semailles ('Harvests and Sowings', 1986), Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner, but which he progressively perceived to be governed by competition and status. He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community. Récoltes et Semailles was finally published in 2022 by Gallimard and, thanks to French science historian Alain Herreman, is also available on the Internet. An English translation by Leila Schneps will be published by MIT Press in 2025. A partial English translation can be found on the Internet. A Japanese translation of the whole book in four volumes was completed by Tsuji Yuichi (1938–2002), a friend of Grothendieck from the Survivre period. The first three volumes (corresponding to Parts 0 to III of the book) were published between 1989 and 1993, while the fourth volume (Part IV) was completed and, although unpublished, copies of it as a typed manuscript are circulated. Grothendieck helped with the translation and wrote a preface for it, in which he called Tsuji his "first true collaborator". Parts of Récoltes et Semailles have been translated into Spanish, as well as into a Russian translation that was published in Moscow. In 1988, Grothendieck declined the Crafoord Prize with an open letter to the media. He wrote that he and other established mathematicians had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community that was characterized by outright scientific theft that he believed had become commonplace and tolerated. The letter also expressed his belief that totally unforeseen events before the end of the century would lead to an unprecedented collapse of civilization. Grothendieck added however that his views were "in no way meant as a criticism of the Royal Academy's aims in the administration of its funds" and he added, "I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy." La Clef des Songes, a 315-page manuscript written in 1987, is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a deity exists. As part of the notes to this manuscript, Grothendieck described the life and the work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age. The only mathematician on his list was Bernhard Riemann. Influenced by the Catholic mystic Marthe Robin who was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988. His growing preoccupation with spiritual matters was also evident in a letter entitled Lettre de la Bonne Nouvelle sent to 250 friends in January 1990. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October 1996. The Grothendieck Festschrift, published in 1990, was a three-volume collection of research papers to mark his sixtieth birthday in 1988. More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal. === Retirement into reclusion and death === In 1991, Grothendieck moved to a new address that he did not share with his previous contacts in the mathematical community. Very few people visited him afterward. Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup. At some point, Leila Schneps and Pierre Lochak located him, then carried on a brief correspondence. Thus they became among "the last members of the mathematical establishment to come into contact with him". After his death, it was revealed that he lived alone in a house in Lasserre, Ariège, a small village at the foot of the Pyrenees. In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de non-publication" to Luc Illusie, claiming that all materials published in his absence had been published without his permission. He asked that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. He characterized a website devoted to his work as "an abomination". His dictate may have been reversed in 2010. In September 2014, almost totally deaf and blind, he asked a neighbour to buy him a revolver so he could kill himself. His neighbour refused to do so. On 13 November 2014, aged 86, Grothendieck died in the hospital of Saint-Lizier or Saint-Girons, Ariège. === Citizenship === Grothendieck was born in Weimar Germany. In 1938, aged ten, he moved to France as a refugee. Records of his nationality were destroyed in the fall of Nazi Germany in 1945 and he did not apply for French citizenship after the war. Thus, he became a stateless person for at least the majority of his working life and he traveled on a Nansen passport. Part of his reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to the Algerian War (1954–62). He eventually applied for French citizenship in the early 1980s, after he was well past the age that would have required him to do military service. === Family === Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 from tuberculosis that she contracted in camps for displaced persons. He had five children: a son with his landlady during his time in Nancy; three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour; and one child with Justine Skalba, with whom he lived in a commune in the early 1970s. == Mathematical work == Grothendieck's early mathematical work was in functional analysis. Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions, and the application of Lp spaces in studying linear maps between topological vector spaces. In a few years, he had become a leading authority on this area of functional analysis—to the extent that Dieudonné compares his impact in this field to that of Banach. It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential work. From approximately 1955 he started to work on sheaf theory and homological algebra, producing the influential "Tôhoku paper" (Sur quelques points d'algèbre homologique, published in the Tohoku Mathematical Journal in 1957) where he introduced abelian categories and applied their theory to show that sheaf cohomology may be defined as certain derived functors in this context. Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray. Grothendieck took them to a higher level of abstraction and turned them into a key organising principle of his theory. He shifted attention from the study of individual varieties to his relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space. In 1956, he applied the same thinking to the Riemann–Roch theorem, which recently had been generalized to any dimension by Hirzebruch. The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand Borel with Serre. This result was his first work in algebraic geometry. Grothendieck went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which at the time were in a state of flux and under discussion in Claude Chevalley's seminar. He outlined his programme in his talk at the 1958 International Congress of Mathematicians. His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closed generic points, which led to the theory of schemes. Grothendieck also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings. His theory of schemes has become established as the best universal foundation for this field, because of its expressiveness as well as its technical depth. In that setting one can use birational geometry, techniques from number theory, Galois theory, commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way. Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal. His influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules. Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. === EGA, SGA, FGA === The bulk of Grothendieck's published work is collected in the monumental, yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). The collection Fondements de la Géometrie Algébrique (FGA), which gathers together talks given in the Séminaire Bourbaki, also contains important material. Grothendieck's work includes the invention of the étale and l-adic cohomology theories, which explain an observation made by André Weil that argued for a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers. Weil had realized that to prove such a connection, one needed a new cohomology theory, but neither he nor any other expert saw how to accomplish this until such a theory was expressed by Grothendieck. This program culminated in the proofs of the Weil conjectures, the last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics. === Major mathematical contributions === In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions that he believed qualified as "great ideas". In chronological order, they are: Topological tensor products and nuclear spaces "Continuous" and "discrete" duality (derived categories, "six operations") Yoga of the Grothendieck–Riemann–Roch theorem K-theory relation with intersection theory Schemes Topoi Étale cohomology and l-adic cohomology Motives and the motivic Galois group (Grothendieck ⊗-categories) Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients"... "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra Tame topology Yoga of anabelian algebraic geometry, Galois–Teichmüller theory "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds Here the term yoga denotes a kind of "meta-theory" that may be used heuristically; Michel Raynaud writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents. Grothendieck wrote that, of these themes, the largest in scope was topoi, as they synthesized algebraic geometry, topology, and arithmetic. The theme that had been most extensively developed was schemes, which were the framework "par excellence" for eight of the other themes (all but 1, 5, and 12). Grothendieck wrote that the first and last themes, topological tensor products and regular configurations, were of more modest size than the others. Topological tensor products had played the role of a tool rather than of a source of inspiration for further developments; but he expected that regular configurations could not be exhausted within the lifetime of a mathematician who devoted oneself to it. He believed that the deepest themes were motives, anabelian geometry, and Galois–Teichmüller theory. == Influence == Grothendieck is considered by many to be the greatest mathematician of the twentieth century. In an obituary David Mumford and John Tate wrote: Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves–and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed. By the 1970s, Grothendieck's work was seen as influential, not only in algebraic geometry and the allied fields of sheaf theory and homological algebra, but influenced logic, in the field of categorical logic. According to mathematician Ravi Vakil, "Whole fields of mathematics speak the language that he set up. We live in this big structure that he built. We take it for granted—the architect is gone". In the same article, Colin McLarty said, "Lots of people today live in Grothendieck's house, unaware that it's Grothendieck's house." === Geometry === Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures. Algebraic geometry has traditionally meant the understanding of geometric objects, such as algebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques of ring theory. In this approach, the properties of a geometric object are related to the properties of an associated ring. The space (e.g., real, complex, or projective) in which the object is defined, is extrinsic to the object, while the ring is intrinsic. Grothendieck laid a new foundation for algebraic geometry by making intrinsic spaces ("spectra") and associated rings the primary objects of study. To that end, he developed the theory of schemes that informally can be thought of as topological spaces on which a commutative ring is associated to every open subset of the space. Schemes have become the basic objects of study for practitioners of modern algebraic geometry. Their use as a foundation allowed geometry to absorb technical advances from other fields. His generalization of the classical Riemann–Roch theorem related topological properties of complex algebraic curves to their algebraic structure and now bears his name, being called "the Grothendieck–Hirzebruch–Riemann–Roch theorem". The tools he developed to prove this theorem started the study of algebraic and topological K-theory, which explores the topological properties of objects by associating them with rings. After direct contact with Grothendieck's ideas at the Bonn Arbeitstagung, topological K-theory was founded by Michael Atiyah and Friedrich Hirzebruch. === Cohomology theories === Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic. The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of the ℓ-adic étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his student Pierre Deligne. Grothendieck's large-scale approach has been called a "visionary program". The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program. Grothendieck's conjectural theory of motives was intended to be the "ℓ-adic" theory but without the choice of "ℓ", a prime number. It did not provide the intended route to the Weil conjectures, but has been behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration. This theory, Daniel Quillen's work, and Grothendieck's theory of Chern classes, are considered the background to the theory of algebraic cobordism, another algebraic analogue of topological ideas. === Category theory === Grothendieck's emphasis on the role of universal properties across varied mathematical structures brought category theory into the mainstream as an organizing principle for mathematics in general. Among its uses, category theory creates a common language for describing similar structures and techniques seen in many different mathematical systems. His notion of abelian category is now the basic object of study in homological algebra. The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, although unintentional. == In popular culture == Colonel Lágrimas (Colonel Tears in English), a novel by Puerto Rican–Costa Rican writer Carlos Fonseca is about Grothendieck. The Benjamín Labatut book When We Cease to Understand the World dedicates one chapter to the work and life of Grothendieck, introducing his story by reference to the Japanese mathematician Shinichi Mochizuki. The book is a lightly fictionalized account of the world of scientific inquiry and was a finalist for the National Book Award. In Cormac McCarthy's The Passenger and its sequel Stella Maris, a main character is a student of Grothendieck's. The Istituto Grothendieck has been created in his honor. == Publications == Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. MR 0075539. OCLC 9308061. Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. == See also == == Notes == == References == == Sources and further reading == == External links == Centre for Grothendieckian Studies (CSG) is a research centre of the Grothendieck Institute, with a dedicated mission to honour the memory of Alexander Grothendieck. O'Connor, John J.; Robertson, Edmund F., "Alexander Grothendieck", MacTutor History of Mathematics Archive, University of St Andrews Alexander Grothendieck at the Mathematics Genealogy Project Séminaire Grothendieck is a peripatetic seminar on Grothendieck view not just on mathematics Grothendieck Circle, collection of mathematical and biographical information, photos, links to his writings The origins of 'Pursuing Stacks': This is an account of how 'Pursuing Stacks' was written in response to a correspondence in English with Ronnie Brown and Tim Porter at Bangor, which continued until 1991. See also Alexander Grothendieck: some recollections. Récoltes et Semailles "Récoltes et Semailles", Spanish translation "La Clef des Songes", French originals and Spanish translations English summary of "La Clef des Songes" Video of a lecture with photos from Grothendieck's life, given by Winfried Scharlau at IHES in 2009 Can one explain schemes to biologists —biographical sketch of Grothendieck by David Mumford & John Tate Archives Grothendieck "Who Is Alexander Grothendieck?, Winfried Scharlau, Notices of the AMS 55(8), 2008. "Alexander Grothendieck: A Country Known Only by Name, Pierre Cartier, Notices of the AMS 62(4), 2015. Alexandre Grothendieck 1928–2014, Part 1, Notices of the AMS 63(3), 2016. Les-archives-insaisissables-d-alexandre-grothendieck Kutateladze S.S. Rebellious Genius: In Memory of Alexander Grothendieck Alexandre-Grothendieck-une-mathematique-en-cathedrale-gothique Les-archives-insaisissables-d-alexandre-grothendieck
Wikipedia:Alexander Kirillov Jr.#0	Alexander Alexandrovich Kirillov Jr. (Russian: Александр Александрович Кириллов) is a Russian-born American mathematician, working in the area of representation theory and Lie groups. He is a son of Russian mathematician Alexandre Kirillov. == Biography == Kirillov received his master's degree from Moscow State University in 1989 and Ph.D from Yale University in 1995. He is currently a professor at State University of New York at Stony Brook. Kirillov is a teacher of the project School Nova, trying to establish traditions of the Russian Mathematical Schools on American soil. He is also a member of the international volunteer organization Rubikus.HelpUA that helps Ukrainian refugees stranded in Russia after the Russian invasion of Ukraine to evacuate to EU. == Publications == Bakalov, Bojko; Kirillov, Alexander Jr. (2001), Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2686-7, MR 1797619 Kirillov, Alexander Jr. (2008), An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, CiteSeerX 10.1.1.173.1452, doi:10.1017/CBO9780511755156, ISBN 978-0-521-88969-8, MR 2440737 Bakalov, Bojko; Kirillov, Alexander Jr. (2016), Quiver Representations and Quiver Varieties, Graduate Studies in Mathematics, vol. 174, Providence, R.I.: American Mathematical Society, ISBN 978-1-4704-2307-0 == References == == External links == Kirillov's homepage Kirillov's page on the SchoolPlus project Alexander Kirillov on the Mathematical Genealogy Project Kirillov's page on the SchoolNova project
Wikipedia:Alexander Kuznetsov (mathematician)#0	Alexander Gennadyevich Kuznetsov (Russian: Александр Геннадьевич Кузнецов, born November 1, 1973) is a Russian mathematician working at the Steklov Mathematical Institute and the Interdisciplinary Scientific Center J.-V. Poncelet, Moscow, and head of the Laboratory of Algebraic Geometry and its Applications of the Higher School of Economics. He graduated from Moscow State School 57 in 1990. He received a Ph.D. in 1998 under the supervision of Alexei Bondal. Kuznetsov is known for his research in algebraic geometry, mostly concerning derived categories of coherent sheaves and their semiorthogonal decompositions. Kuznetsov received an August Möbius fellowship in 1997. He was awarded a European Mathematical Society prize in 2008. He was an invited speaker at the International Mathematical Congress in Seoul (2014). Kuznetsov is a Professor of the Russian Academy of Sciences (RAS), corresponding member of the RAS (elected in 2016). == Selected publications == Kuznetsov, Alexander (2007). "Homological projective duality". Publications Mathématiques de l'IHÉS. 105 (1): 157–220. arXiv:math/0507292. doi:10.1007/s10240-007-0006-8. Kuznetsov, Alexander (2011). "Base change for semiorthogonal decompositions". Compositio Mathematica. 147 (3): 852–876. arXiv:0711.1734. doi:10.1112/S0010437X10005166. Kuznetsov, A. G. (2009). "Derived categories of Fano threefolds". Proceedings of the Steklov Institute of Mathematics. 264: 110–122. arXiv:0809.0225. doi:10.1134/S0081543809010143. Kuznetsov, Alexander (2008). "Derived categories of quadric fibrations and intersections of quadrics". Advances in Mathematics. 218 (5): 1340–1369. arXiv:math/0510670. doi:10.1016/j.aim.2008.03.007. == References == == External links == Official home page Google scholar profile Math-Net.ru profile MathSciNet profile
Wikipedia:Alexander Merkurjev#0	Aleksandr Sergeyevich Merkurjev (Russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles. == Work == Merkurjev's work focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K-theory and central simple algebras. In the early 1980s Merkurjev proved a fundamental result about the structure of central simple algebras of period dividing 2, which relates the 2-torsion of the Brauer group with Milnor K-theory. In subsequent work with Suslin this was extended to higher torsion as the Merkurjev–Suslin theorem. The full statement of the norm residue isomorphism theorem (also known as the Bloch-Kato conjecture) was proven by Voevodsky. In the late 1990s Merkurjev gave the most general approach to the notion of essential dimension, introduced by Buhler and Reichstein, and made fundamental contributions to that field. In particular Merkurjev determined the essential p-dimension of central simple algebras of degree p 2 {\displaystyle p^{2}} (for a prime p) and, in joint work with Karpenko, the essential dimension of finite p-groups. == Awards == Merkurjev won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on algebraic K-theory. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley, California, and his talk was entitled "Milnor K-theory and Galois cohomology". In 1995 he won the Humboldt Prize, an international prize awarded to renowned scholars. Merkurjev gave a plenary talk at the second European Congress of Mathematics in Budapest, Hungary in 1996. In 2012 he won the Cole Prize in Algebra for his work on the essential dimension of groups. In 2015 a special volume of Documenta Mathematica was published in honor of Merkurjev's sixtieth birthday. == Bibliography == === Books === Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol: The book of involutions, American Mathematical Society 1998. ISBN 0-8218-0904-0 Skip Garibaldi, Jean-Pierre Serre, Alexander Merkurjev: Cohomological Invariants in Galois Cohomology, American Mathematical Society 2003. ISBN 0-8218-3287-5 Richard Elman, Nikita Karpenko, Alexander Merkurjev: Algebraic and geometric theory of quadratic forms, American Mathematical Society 2008. ISBN 978-0-8218-4329-1 == References == == External links == Alexander Merkurjev - personal webpage at UCLA Alexander Merkurjev at the Mathematics Genealogy Project Alexander Merkurjev's results at International Mathematical Olympiad
Wikipedia:Alexander Mielke#0	Alexander Mielke (born 14 September 1958.) is a German mathematician working in the areas of nonlinear partial differential equations and applied analysis. He is a professor of applied analysis at the Humboldt University of Berlin and heads the research group on partial differential equations at the Weierstrass Institute == Education and Career == Mielke received his PhD from the University of Stuttgart in 1984 under the supervision of Klaus Kirchgässner, and his thesis was titled Stationary Solutions of the Euler Equation in Channels of Variable Depth.. He was a postdoctoral researcher at Cornell University under the supervision of Philip Holmes between 1986 and 1987, and in 1990, he habilitated at the University of Stuttgart on the topic of Hamiltonian and Lagrangian flows on center manifolds with applications to elliptic variational problems and was appointed professor at the University of Hannover in 1992. In 1999, he was appointed at the Institute for Analysis, Dynamics, and Modeling at the University of Stuttgart. He has been a full professor at the Humboldt University of Berlin since 2004 and Head of the Research Group Partial differential equations at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. === Awards and Honours === Richard von Mises Prize of the Society of Applied Mathematics and Mechanics (GAMM), 1989. Heinz Maier-Leibnitz-Preis awarded by the German Research Foundation, 1989. ISIMM Senior prize, 2024. == References ==
Wikipedia:Alexander Moiseevich Olevskii#0	Alexander Moiseevich Olevskii (Russian: Александр Моисеевич Олевский, born February 12, 1939, in Moscow) is a Russian-Israeli mathematician at Tel Aviv University, specializing in mathematical analysis. As of July 2021, he is a professor emeritus. He graduated in 1963 with a Candidate of Sciences degree (PhD) from Moscow State University. There he received in 1966 a Russian Doctor of Sciences degree (habilitation). At the Moscow Institute of Electronics and Mathematics, he was from 1988 to 1992 head of the department of algebra and analysis. In the spring of 1996 he was at the Institute for Advanced Study. He has held visiting appointments at universities or institutes in several countries, including France, Australia, Germany, Italy, and the United States. In 1986 Olevskii was an invited speaker at the International Congress of Mathematicians in Berkeley, California. He was a member of the 2013 Class of Fellows of the American Mathematical Society (announced in 2012). In 2014 he was an invited speaker at the European Congress of Mathematics in Kraków. His doctoral students include Gady Kozma. == Selected publications == Olevskii, A. M. (1969). "On the extension of a sequence of functions to a complete orthonormal system". Mathematical Notes of the Academy of Sciences of the USSR. 6 (6): 908–913. doi:10.1007/BF01146412. S2CID 120976873. Olevskii, A. M. (1970). "Stability of the Schmidt orthogonalization operator". Izv. Akad. Nauk SSSR Ser. Mat. 34 (4): 803–826. Bibcode:1970IzMat...4..811O. doi:10.1070/IM1970v004n04ABEH000934. Olevskii, A. M. (1972). "On operators generating conditional bases in a Hilbert space". Mathematical Notes of the Academy of Sciences of the USSR. 12: 476–482. doi:10.1007/BF01094395. S2CID 124705771. Olevskii, A. M. (1975). "Fourier Series with Respect to General Orthogonal Systems". Ergebn. Math. Grenzgen. 86: 1–136. ISBN 3540071032.} Olevskii, A. M. (1985). "Modifications of functions and Fourier series". Russian Mathematical Surveys. 40 (3): 181. Bibcode:1985RuMaS..40..181O. doi:10.1070/RM1985v040n03ABEH003592. S2CID 250889771. Olevskii, A. M. (2004). "Representation of functions by exponentials with positive frequencies". Russian Mathematical Surveys. 59 (1): 171–180. Bibcode:2004RuMaS..59..171O. doi:10.1070/RM2004v059n01ABEH000707. S2CID 250891540. 2004 Lebedev, V. V.; Olevskii, A. M. (2006). " L p {\displaystyle L^{p}} -Fourier multipliers with bounded powers". Izv. RAN. Ser. Mat. 7 (3): 129–166. Bibcode:2006IzMat..70..549L. doi:10.1070/IM2006v070n03ABEH002319. S2CID 250919411. Olevskii, Alexander M.; Ulanovskii, Alexander (13 June 2016). Functions with Disconnected Spectrum. American Mathematical Soc. ISBN 9781470428891. == References == == External links == "Olevskii, Alexander Moiseevich". Math-Net.Ru. (publication list) "Лекция 1 \| Квазикристаллы и формула суммирования Пуассона \| Александр Олевский \| Лекториум". YouTube. Лекториум. May 17, 2015.
Wikipedia:Alexander Molev#0	Alexander Ivanovich Molev (Russian: Алекса́ндр Ива́нович Мо́лев) (born 1961) is a Russian-Australian mathematician. He completed his Ph.D. in 1986 under the supervision of Alexandre Kirillov at Moscow State University. He was awarded the Australian Mathematical Society Medal in 2001 and became a Fellow of the Australian Academy of Science in 2019. Amongst other things, he has worked on Yangians and Lie algebras. He is currently a Professor in the School of Mathematics and Statistics, Faculty of Science, University of Sydney. == Bibliography == Alexander Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, 143. American Mathematical Society, Providence, RI, 2007. xviii+400 pp. ISBN 978-0-8218-4374-1 Alexander Molev, Sugawara Operators for Classical Lie Algebras, Mathematical Surveys and Monographs, 229. American Mathematical Society, Providence, RI, 2018 304 pp. ISBN 978-1-4704-3659-9 == External links == University home page
Wikipedia:Alexander Myller#0	Alexander Myller (1879–1965) was a Romanian mathematician and professor at the Alexandru Ioan Cuza University of Iași. == References ==
Wikipedia:Alexander Nabutovsky#0	Alexander Nabutovsky is a leading Canadian mathematician specializing in differential geometry, geometric calculus of variations and quantitative aspects of topology of manifolds. He is a professor at the University of Toronto Department of Mathematics. Nabutovsky earned a Ph.D. degree from the Weizmann Institute of Science in 1993; his advisor was Shmuel Kiro. He was an invited speaker on "Geometry" at International Congress of Mathematicians, 2010 in Hyderabad. == References == == External links == Alexander Nabutovsky. "Publication list". personal webpage at the University of Toronto.
Wikipedia:Alexander Olshanskii#0	Alexander Yu. Olshanskii (Russian: Александр Юрьевич Ольшанский; born 19 January 1946) is a Russian-American mathematician renowned for his contributions to combinatorial and geometric group theory. He is particularly noted for constructing infinite groups with unusual properties and for developing geometric methods in group theory. Olshanskii has also made significant contributions to the study of Lie algebras and algebraic systems. He served as the Centennial Professor of Mathematics at Vanderbilt University from 1999 until becoming Professor Emeritus in 2024. == Early Life and Education == Olshanskii was born in Saratov, Russia, into a family of a military aviation engineer and is one of three brothers. He completed high school in Engels in 1963 and graduated from Moscow State University in 1968 with a B.S. in Mathematics. He earned his Ph.D. in Mathematics in 1971 under the supervision of Alfred Lvovich Shmelkin, with a dissertation titled "Some Questions of the Theory of Group Varieties." He later received his Doctor of Sciences degree (Habilitation) in 1979. == Academic Career == From 1970, Olshanskii was a faculty member in the Department of Mechanics and Mathematics at Moscow State University, becoming an associate professor in 1978 and a full professor in 1985. In 1983, he was an invited speaker at the International Congress of Mathematicians in Warsaw, where he presented on geometric methods in combinatorial group theory. In 1999, he joined Vanderbilt University as the Centennial Professor of Mathematics. He was named a Fellow of the American Mathematical Society in 2014 for his contributions to group theory. He became Professor Emeritus at Vanderbilt University in 2024. == Research Contributions == Olshanskii's research has significantly influenced combinatorial and geometric group theory. In 1969, while still a graduate student, he resolved a problem posed by Bernard Neumann in 1935 concerning the existence of an infinite system of group identities not equivalent to any finite system. This achievement garnered recognition from Neumann, who was then at Vanderbilt University. In the late 1970s and early 1980s, Olshanskii introduced graded van Kampen diagrams, a refinement of the classical van Kampen diagrams. Using this technique, he constructed the so-called Tarski monster groups: infinite groups of bounded exponent in which every proper subgroup is cyclic. These constructions provided counterexamples to several longstanding conjectures in group theory, including the von Neumann conjecture. In 1980, he constructed a counterexample to the von Neumann–Day problem, demonstrating the existence of non-amenable groups without non-cyclic free subgroups. Olshanskii also provided a new geometric proof of the Novikov--Adyan theorems addressing the Burnside problem. His proof for large odd exponents was notably concise—32 pages compared to the original's 300—and is still considered the most succinct. His work has influenced the development of hyperbolic groups, a concept central to geometric group theory. Olshanskii extended his methods to study small cancellation theory and van Kampen diagrams over hyperbolic groups, and investigated factor groups of such groups. In recent years, Olshanskii has focused on asymptotic invariants of groups, such as Dehn functions, distortion, and relative subgroup growth. These invariants are connected to the complexity of algorithmic problems in group theory. In collaboration with M. Birget, E. Rips, and M. Sapir, he developed a geometric criterion for determining when the word problem in a finitely presented group can be solved in nondeterministic polynomial time. Olshanskii has authored over 100 scientific papers, including the influential monograph Geometry of Defining Relations in Groups. == Honors and Awards == Invited Speaker at the International Congress of Mathematicians (1983) Maltsev Prize of the Russian Academy of Sciences (2000) Fellow of the American Mathematical Society (2014) The Moscow Mathematical Society award (1969) == References == == External links == Vanderbilt University Profile Alexander Olshanskii – Mathematics Genealogy Project Alexander Olshanskii – Google Scholar Александр Юрьевич Ольшанский on halgebra.math.msu.su Александр Юрьевич Ольшанский on istina.msu.ru
Wikipedia:Alexander Ostrowski#0	Alexander Markowich Ostrowski (Ukrainian: Олександр Маркович Островський; Russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893 – 20 November 1986) was a mathematician. == Biography == His father Mark having been a merchant, Alexander Ostrowski attended the Kiev College of Commerce, not a high school, and thus had an insufficient qualification to be admitted to university. However, his talent did not remain undetected: Ostrowski's mentor, Dmitry Grave, wrote to Edmund Landau and Kurt Hensel for help. Subsequently, Ostrowski began to study mathematics at Marburg University under Hensel's supervision in 1912. During World War I he was interned, but thanks to the intervention of Hensel, the restrictions on his movements were eased somewhat, and he was allowed to use the university library. After the war ended, Ostrowski moved to Göttingen where he wrote his doctoral dissertation and was influenced by David Hilbert, Felix Klein, and Landau. In 1920, after having obtained his doctorate from the University of Göttingen, Ostrowski moved to Hamburg where he worked as Erich Hecke's assistant and finished his habilitation in 1922. In 1923, he returned to Göttingen and, in 1928, became Professor of Mathematics at the University of Basel, until retirement in 1958. In 1950, Ostrowski obtained Swiss citizenship. After retirement, he still published scientific papers until his late eighties. His work was mainly, but not exclusively in algebra and number theory, and he had a great interest in numerical methods as well as abstract mathematics. == Selected publications == Vorlesungen über Differential- und Integralrechnung, 3 vols., Birkhäuser; vol. 1, 1945; vol. 1, 2nd edition, 1960; vol. 2, 1951; vol. 3, 1954; Solution of equations and systems of equations. Academic Press, New York 1960; 2nd edition 1965; 2016 pbk reprint of 2nd edition Aufgabensammlung zur Infinitesimalrechnung. several vols., Birkhäuser, Basel (1st edition 1964; 2nd edition 1972) pbk reprint vol. 1; vol. 2 A; vol. 2 B; vol. 3 Collected mathematical papers. 6 vols., Birkhäuser, Basel 1983–1984. vol. 1; vol. 2; vol. 3; vol. 4; vol. 5; vol. 6 == See also == Ostrowski's theorem Ostrowski–Hadamard gap theorem Ostrowski numeration Ostrowski Prize == References == == External links == Ostrowski Foundation website: short biography O'Connor, John J.; Robertson, Edmund F., "Alexander Ostrowski", MacTutor History of Mathematics Archive, University of St Andrews Gautschi, Walter, Alexander M. Ostrowski (1893-1986): His life, work, and students
Wikipedia:Alexander Provan Robertson#0	Alexander Provan Robertson FRSE FIMA (16 June 1925 – 31 January 1995) was a 20th century Scottish mathematician who emigrated to Australia. == Life == He was born on 16 June 1925 in Glasgow the only child of an assistant railway station master. He was educated at Shawlands Academy and won a bursary to Glasgow University in 1942, graduating MA in 1946. After 18 months assisting in lectures in Mathematics he won the Ferguson Scholarship and went to Cambridge University to begin the Mathematics Tripos, gaining a BA in 1951 and doctorate. At Cambridge he also enjoyed organ-playing under Dr George Guest. In 1952 he returned to Glasgow University as a Senior Lecturer in Mathematics. Around 1956 he was elected a Fellow of the Royal Society of Edinburgh. He resigned in 1989. In 1963 he was given the Chair in Mathematics at Keele University. In 1973 he emigrated with his family to Perth, Western Australia to take a professorship at Murdoch University and to help in its foundation. He retired in 1990 and died on 31 January 1995. == Family == He was married to Wendy Sadie in 1951. == Publications == Topological Vector Spaces (1973) == References ==
Wikipedia:Alexander Ramm#0	Alexander G. Ramm (born 1940 in St. Petersburg, Russia) is an American mathematician. His research focuses on differential and integral equations, operator theory, ill-posed and inverse problems, scattering theory, functional analysis, spectral theory, numerical analysis, theoretical electrical engineering, signal estimation, and tomography. == Education and career == Ramm obtained a B.S. degree in mathematics in 1959 and an M.S. degree in 1961 both at Leningrad State University. He received a Ph.D. degree from Moscow State University in 1964 and Dr. Sci. in 1972 at the Mathematics Institute Academy of Science, Minsk. Ramm taught at Leningrad Institute of Precision Mechanics and Optics from 1962 to 1979. He was a visiting professor and research scientist at the University of Michigan in 1979–1981. He has been a professor at Kansas State University since 1981, and lectured at many universities and research centers around the world. He is currently professor emeritus. == Awards and honors == Ramm received Distinguished Graduate faculty award in 1996 and received Khwarizmi International Award for mathematical research in 2004. He was a distinguished foreign professor at the Academy of Science of Mexico (1997), research CNRS professor in France (2003), distinguished visiting professor at the University of Cairo (2004, 2006), distinguished visiting professor supported by the UK Royal Academy of Engineering (2009). He was a Mercator Professor in 2007, Distinguished HKSTAM speaker (2005), London Mathematical Society speaker (2005). Ramm was a Fulbright Research Professor in Israel (Technion) in 1991–1992, an invited plenary speaker at 7th PACOM in 2009. He was a visiting professor at IMPAN in 2010, at MPI (Max Planck Institute) in 2011, at Beijing Institute of Technology (BIT) in 2013, a Fulbright Research Professor at the University of Lviv, Ukraine, in 2015. Ramm was an elected member of Electromagnetic Academy, MIT (June 1990) and a member of New York Academy of Science. He has been an associated editor of many professional journals. == Research == Ramm's work can be divided into the following areas: PDE, ODE and integral equations, spectral and scattering theory for differential operators, especially for Schrödinger operators, static problems and wave scattering by small bodies of arbitrary shapes, random fields estimation theory, nonlinear passive systems, inverse scattering problems theoretical numerical analysis and ill-posed problems, nonselfadjoint operators and their applications in scattering theory, signal and image processing, local tomography, mathematical geophysics, electromagnetic theory and mathematical physics, creating materials with a desired refraction coefficient, symmetry problems for PDE, Navier-Stokes problem in R 3 {\displaystyle \mathbb {R} ^{3}} , inverse scattering with non-over-determined scattering data. Highlights of Ramm's research are: In a long series of papers starting with a thorough study of the spectral properties and eigenfunction expansions is given for the first time for Schrödinger operators in domains with infinite boundaries; Iterative methods are developed for solving interior and exterior boundary value problems for Laplace's equation, analytic formulas for the S-matrix for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes are derived and applied successfully to numerical and physical problems (see ); Analytic theory of random fields estimation was developed in monograph which is an original detailed study of a new class of multidimensional integral equations basic in estimation theory. No results of this type had been known prior the work of Ramm. This monograph was translated into Russian by MIR publishing house in 1996. Many results known for one-dimensional estimation theory are very particular cases of the general theory developed in the monograph. The theory has many applications in signal processing, and in geophysics in particular. In the papers and (also,) the mathematical foundations of the EEM and SEM methods are given. These methods are now very popular in electrical engineering sciences. A thorough study of existence, global stability and calculation of the stationary regimes in passive nonlinear systems is given in paper. The results are optimal as shown by examples. A study of inverse scattering problems is given in a long series of papers (see monographs, and papers,) where a summary of some of the author's results is given. In a recent paper the problem which has been open for many decades is solved: uniqueness of the solution to a non-overdetermined inverse scattering problem is proved. Exact inversion of low-frequency scattering data is given in the book. A powerful method, Property C method, based on the notion of completeness of the set of products of solutions of PDE is developed and applied to many important inverse problems. In these works several problems are solved which have been open for decades. For example, the first global uniqueness theorems in geophysics and potential scattering with fixed-energy data are obtained, the first mathematically justified method for solving the 3D inverse scattering problem with noisy fixed-energy data is given, and, for the first time, stability estimates for the solution to the inverse scattering problem with noisy fixed-energy data are obtained.The first variational principle for solving inverse scattering problems which is equivalent to the inverse problems was found; this work is published as a monograph, which is an expanded version of monograph, translated into Russian in 1994. Very recently (paper ) a fundamentally new uniqueness theorem is obtained: it says that a compactly supported real-valued square-integrable spherically symmetric potential is uniquely defined by any part of the fixed-energy phase shifts with the angular momenta j {\displaystyle j} running through an arbitrary set J {\displaystyle J} of non-negative integers such that ∑ j ∈ J , j ≠ 0 1 j = ∞ {\displaystyle \sum _{j\in J,j\not =0}{\frac {1}{j}}=\infty } . Property C is defined and proved for ordinary differential equations (ODE) and its many new applications are demonstrated. Most of the known results for one-dimensional inverse problems are obtained by using this property, and many new results. Among the classical results which are obtained by using property C for ODE are Marchenko and Borg's uniqueness theorems concerning recovery of the potential from two spectra and from scattering data or spectral function. Inverse problems for an inhomogeneous Schrödinger equation are studied for the first time, a non-over-determined three-dimensional inverse problem of recovery of a potential from the diagonal values of the spectral function known on the boundary of a bounded domain and all real values of the spectral parameter is considered and a uniqueness theorem is proved for this problem. A new approximate method for solving the inverse scattering problem with fixed energy data is given for a spherically symmetric potentials which are known for r > a but unknown for r < a {\displaystyle r<a} , where a > 0 {\displaystyle a>0} is an arbitrary large fixed number. Numerical results are obtained by this method. Krein's method in inverse scattering is justified and its consistency is proved. Analytical theory is given for inversion of the surface scattering data in the ground-penetrating radar problem for two functions: permittivity and conductivity of the ground, under the assumption that these functions depend on the vertical coordinate only,. A method for recovery of a quarkonium system from experimental data is developed. Inverse problem of finding point scatterers from the surface scattering data is posed and solved. For the first time uniqueness theorems are proved for three-dimensional scattering problems with non-overdetermined data. Stability of the Pompeiu property is established and further results are obtained. In papers and a method for constructing a ”smart material” is given. It is proved that one can distribute small particles in a bounded domain so that the resulting material has the a priori chosen radiation pattern. Moreover, a method for calculating the density of these particles and their properties is developed. In paper theory of scalar wave scattering by one and many small bodies of an arbitrary shape is developed for various boundary conditions (Dirichlet, Neumann, impedance, transmission). In paper theory of EM (electromagnetic) wave scattering by one and many small impedance bodies of an arbitrary shape is developed. Methods for creating materials with a desired refraction coefficient are given on the basis of the above theory. In and in monographs, for the first time the 3D inverse scattering problem is studied for the non-overdetermined scattering data and the uniqueness of its solution is proved. Mathematical justification of the T-matrix approach in scattering theory is given. In a series of papers several ill-posed problems are investigated. In particular, the now widely used stable differentiation procedure based on the regularization by the choice of the step size in the divided difference formula has been introduced originally in. The important feature of this and other of my works on ill-posed problems is the error estimates with explicitly written estimation constants. A theory for stable solution of a class of Fredholm equations at a characteristic value is constructed in several papers and presented systematically in the monograph. This theory was a basis for the theory of wave scattering by small bodies of arbitrary shapes in this monograph. Numerical methods were given for solving integral equations of estimation theory in distributions. This theory is summarized in the monograph. The basis of it is a theory, developed by the author, of a class of multidimensional integral equations whose kernels are kernels of positive rational functions of arbitrary selfadjoint elliptic operators. In a series of papers some of which are joint with Ramm's Ph.D. students, and in monograph a general method, the Dynamical Systems Method (DSM) for treating linear and, especially, nonlinear ill-posed problems by solving a suitable Cauchy problem in a Hilbert space was developed. Convergence theorems are proved. Discretization of the Cauchy problem leads to a variety of iterative methods for solving ill-posed nonlinear problems and convergence theorems for these methods are obtained. In monograph these results are illustrated by numerical examples. A novel approach to solving exterior and interior boundary value problems and scattering problems, based on the theorem, proved by Ramm and called Modified Rayleigh conjecture, has been developed and tested numerically (papers,). The theory of weakly non-selfadjoint operators was applied to scattering theory. For the first time completeness of the set of root vectors of some non-selfadjoint integral operators arising in diffraction and scattering theory was proved. This gave a mathematical justification of the EEM (eigenmode expansion method), a popular method in electrical engineering. Jointly with his Ph.D. student A. I. Katsevich, numerical methods for signal and image processing, edge detection in particular, are developed, and a very general test of randomness against fairly broad alternatives is found and justified mathematically. New methods were developed jointly with A. I. Katsevich for finding jumps of functions from local tomographic data. These methods turned to be practically important. These results were tested numerically and practically and demonstrated their effectiveness. A monograph () contained these results was published in 1996 jointly with A. I. Katsevich. Two patents (5,539,800 of July 23, 1996 and 5,550,892 of Aug. 27, 1996) have been issued by the US Patent Office to A. G. Ramm and A. I. Katsevich ”Enhanced local tomography” and ”Pseudolocal tomography”. A systematic study of the singularities of the Radon transform is given, a complete description of the asymptotics of the Radon transform near a point of its singular support is obtained and applied to the important problem of tomography: finding singularities of a function from its tomographic data; these results are published in a series of papers and appeared in the monograph. The uniqueness theorems for model inverse problems of geophysics have been proved, examples of non-uniqueness were constructed, the theory of inversion of low-frequency data has been developed (monographs and ). Theoretical investigation of a number of antenna synthesis problems, including a non-linear synthesis problems have been investigated. Degree of non-uniqueness of the solution to the general synthesis problem has been described (monograph,). There are many other results of various nature and in different branches of mathematics: general relativity, asymptotic of the spectra of linear operators and quadratic forms, approximation theory, variational estimates of capacitances and polarizabilities, methods for calculation of resonances in open systems and quantum mechanics, perturbation theory for resonances, impedance tomography, singular perturbation of integral equations, quantum chaos, etc. The characteristic features of the works is a systematic usage of functional analysis and classical analysis, numerical methods, PDE, physics and theoretical engineering and their combinations. Broad interests made it possible to interact with mathematicians and engineers with quite diverse interests. In 2007-2017 A.G.Ramm has published a series of papers (-,-,-, and in monographs and ) in which he has developed a method for creating materials with a desired refraction coefficient. This method is based on Ramm's solution to many-body scattering problem by many small particles embedded in an inhomogeneous medium. The refraction coefficient can be created so that the new material has a desired wave-focusing property, or it may have a negative refraction property, which means that the group velocity in this material is directed opposite to the phase velocity. These results are presented in monographs and. Monograph is the second edition of. These results will be immediately applicable practically if small impedance particles with a desired refraction coefficient can be produced in practice. In 2017-2019 A.G. Ramm was working on symmetry problems for PDE. His new results, including the proof of the Schiffer's conjecture and a solution to the Pompeiu problem are presented in the monograph and in papers. In 2019-2021 A.G.Ramm worked on the Navier-Stokes problem (NSP). He published monograph R707 where a detailed analysis of the NSP is given. It is proved that the Navier-Stokes equations are contradictory. In paper the NSP paradox is formulated. These results solve the millennium Navier-Stokes problem in R 3 {\displaystyle \mathbb {R} ^{3}} . As of August 24, 2022, this solution has not been accepted by the Clay Mathematics Institute, but nobody so far pointed any unclear points in the monograph. There is a review of the paper published in Zentralblatt, where an estimate in is claimed to be wrong. This claim in Zentralblatt review is erroneous as was proved in. In 2017-2019 A.G.Ramm has proved for the first time uniqueness of the solution to the inverse scattering problem for compactly supported potentials and non-over-determined scattering data. These results are published in monograph and in the author's papers cited there, in particular, in,. His theory includes a proof of uniqueness of the solution to inverse obstacle scattering problem with non-over-determined data. These results are presented in papers, and in monographs. In 2018–2022 A. G. Ramm has developed a theory for solving convolution integral equations with highly singular kernels. In 2022 A.G. Ramm proved existence and uniqueness of the solution to the Dirichlet problem with $L^1(S)$ boundary data. In, global existence and stability of solutions to nonlinear differential equations are presented. In the size of a scatterer is estimated from its scattering amplitude. == References ==
Wikipedia:Alexander Razgulin#0	Alexander Razgulin (Russian: Алекса́ндр Вита́льевич Разгу́лин) (born 1963) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University. He defended the thesis «Stable method for solving linear equations with noncompact operators and its applications to control and observation problems» for the degree of Doctor of Physical and Mathematical Sciences (2009). Author of 24 books and more than 100 scientific articles. == References == == Bibliography == Grigoriev, Evgeny (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory. Moscow: Publishing house of Moscow University. pp. 50–51. ISBN 978-5-211-05838-5. == External links == MSU CMC(in Russian) Scientific works of Alexander Razgulin Scientific works of Alexander Razgulin(in English)
Wikipedia:Alexander Samarskii#0	Alexander Andreyevich Samarskii (Russian: Александр Андреевич Самарский; 19 February 1919, Amvrosiivka, metropolitan Donetsk, Yekaterinoslav Governorate – 11 February 2008, Moscow) was a Soviet and Russian mathematician and academician (Academy of Sciences of the Soviet Union, Russian Academy of Sciences), specializing in mathematical physics, applied mathematics, numerical analysis, mathematical modeling, finite difference methods. == Education and career == Born in Amvrosiivka, Yekaterinoslav Governorate, Russian Empire (now, Donetsk Oblast, Ukraine). Samarskii studied from 1936 at Moscow State University, interrupted from 1941 to 1944 by voluntary military service in WW II — he was severely wounded in the Battle of Moscow. In 1948 he received his Russian candidate degree (Ph.D.). At the same time, he worked with Andrey Tikhonov on mathematical modeling of nuclear weapon explosions and electromagnetic fields in waveguides. In the 1950s Samarskii worked on finite difference methods and became the founder of a Soviet school in this field. In 1957 he received his Russian doctorate (higher doctoral degree) at the Keldysh Institute of Applied Mathematics with a dissertation on the solution of nonlinear problems of mathematics using finite difference methods. He was elected in 1966 a corresponding member and in 1976 a full member of the Russian Academy of Sciences. He was in 1953 a department head at the Institute of Applied Mathematics of the Soviet Academy of Sciences and a professor at Moscow State University, where he also received an honorary professorship. He founded the Department of Computational Modeling in the Faculty of Numerical Mathematics of Moscow State University and the Department of Mathematical Modelling at the Moscow Institute of Physics and Technology. From 1991 to 1998 he headed the Institute of Mathematical Modelling of the Russian Mathematical Modeling Committee (the Russian part of the International Association for Mathematics and Computers in Simulation (IMACS)). Samarskii received numerous honors and awards for his work, including the Hero of Socialist Labour (1979) and received the Lenin Prize (1962), the USSR State Prize (1965), the State Prize of the Russian Federation (1999), and the Lomonosov Prize of Moscow State University (1997). He was an Invited Speaker of the ICM in Vancouver in 1974 and in 1978 in Helsinki. He was elected in 1985 a foreign member of the Akademie der Wissenschaften der DDR, the National Academy of Sciences of Belarus, and the National Academy of Sciences of Ukraine. == Research == Samarskii is the founder of the national mathematical modeling, the leading expert in the field of computational mathematics, mathematical physics, theory of difference schemes, numerical simulation of complex nonlinear systems. He is the creator of the theory of operator-difference schemes, general theory of stability of difference schemes. Samarskii achieved fundamental results in finite difference methods, for example, finite difference methods applied to stability theory, the application of grid approximations to equations of mathematical physics (including non-classical problems and non-linear equations such as in chaotic diffusion), and methods for solving lattice equations. Samarskii and his students developed analytic and numerical methods for solving problems in nuclear physics, plasma physics, nuclear fusion, magnetohydrodynamics, gas dynamics, hydrodynamics with radiation interaction, laser thermochemistry, convection, ecology, and autocatalytic chemical reactions. Samarskii had over 100 doctoral students and his students include three academicians and five corresponding members of the Russian Academy of Sciences. == Selected publication == Theorie der Differenzenverfahren. Leipzig, 1984, Academische Verlagsgessellschaft, 356 p. The theory of difference schemes. New York – Basel. Marcel Dekker, Inc, 2001, pp. 761. with A. N. Tikhonov: Differentialgleichungen der Mathematischen Physik (series Hochschulbücher für Mathematik. vol. 39). Deutscher Verlag der Wissenschaften, Berlin 1959, English transliteration: Equations of Mathematical Physics, Pergamon Press, Oxford-London-NY-Paris, 1963, Dover 1990 with B. M. Budak, A. N. Tikhonov: A collection of problems of mathematical physics. Pergamon Press 1964, New York, Dover Publications. Inc., 1988, 768 pp. ISBN 0-486-65806-6 with A. N. Tikhonov: Partial differential equations of mathematical physics. 2 vols. Holden-Day, San Francisco 1964, 1967. with Evgenii S. Nikolaev: Numerical methods for grid equations. 2 vols. Birkhäuser, 1989. with Galactionov V.A and co.Blow-up in quasilinear parabolic equations. Walter de Gruyte Berlin, NY, 1995, 534 p. ISBN 3-11- 012754-7. with P.N. Vabishchevich: Computational heat transfer. 2 vols. Chichester, Wiley. 1995. with A. P. Mikhailov: Principles of mathematical modeling: ideas, methods and examples. London and New York. Taylor and Francis, 2002, 349 c. ISBN 0 -415-27280-7. with P. P. Matus, P.N. Vabishchevich: Difference methods with operator factors. Kluwer Academic Publishers, with P.N. Vabishchevich:Numerical methods for solving inverse problems of mathematical physics. Walter de Gruyte Berlin, NY de Gruyter, 2007. == References == == External links == Literature by and about Alexander Samarskii in the German National Library catalogue Samarski on mathnet.ru Alexander Andreyevich Samarskii's books, MacTutor History "Samarskii Alexander Andreyevich". Memorial Site – М., 2013. Retrieved 2018-02-19.
Wikipedia:Alexander Schrijver#0	Alexander (Lex) Schrijver (born 4 May 1948 in Amsterdam) is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in Amsterdam. Since 1993 he has been co-editor in chief of the journal Combinatorica. == Biography == Schrijver earned his Ph.D. in 1977 from the Vrije Universiteit in Amsterdam, under the supervision of Pieter Cornelis Baayen. He worked for the Centrum Wiskunde & Informatica (under its former name as the Mathematisch Centrum) in pure mathematics from 1973 to 1979, and was a professor at Tilburg University from 1983 to 1989. In 1989 he rejoined the Centrum Wiskunde & Informatica, and in 1990 he also became a professor at the University of Amsterdam. In 2005, he stepped down from management at CWI and instead became a CWI Fellow. == Awards and honors == Schrijver was one of the winners of the Delbert Ray Fulkerson Prize of the American Mathematical Society in 1982 for his work with Martin Grötschel and László Lovász on applications of the ellipsoid method to combinatorial optimization; he won the same prize in 2003 (shared with Satoru Iwata, Lisa Fleischer, and Satoru Fujishige) for showing submodular minimization to be strongly polynomial. He won the INFORMS Frederick W. Lanchester Prize in 1986 for his book Theory of Linear and Integer Programming, and again in 2004 for his book Combinatorial Optimization: Polyhedra and Efficiency. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1986 in Berkeley and of the ICM in 1998 in Berlin. In 2003, he won the George B. Dantzig Prize of the Mathematical Programming Society and SIAM for "deep and fundamental research contributions to discrete optimization". In 2006, he was a joint winner of the INFORMS John von Neumann Theory Prize with Grötschel and Lovász for their work in combinatorial optimization, and in particular for their joint work in the book Geometric Algorithms and Combinatorial Optimization showing the polynomial-time equivalence of separation and optimization. In 2008, his work with Adri Steenbeek on scheduling the Dutch train system was honored with INFORMS' Franz Edelman Award for Achievement in Operations Research and the Management Sciences. He won the SIGMA prize of the Dutch SURF foundation in 2008, for a mathematics education project. In 2015 he won the EURO Gold Medal, the highest distinction within Operations Research in Europe. In 2005 Schrijver won the Spinoza Prize of the NWO, the highest scientific award in the Netherlands, for his research in combinatorics and algorithms. Later in the same year he became a Knight of the Order of the Netherlands Lion. In 2002, Schrijver received an honorary doctorate from the University of Waterloo in Canada, and in 2011 he received another one from Eötvös Loránd University in Hungary. Schrijver became a member of the Royal Netherlands Academy of Arts and Sciences in 1995. He became a corresponding member of the North Rhine-Westphalia Academy for Sciences and Arts in 2005, joined the German Academy of Sciences Leopoldina in 2006, and was elected to the Academia Europaea in 2008. In 2012 he became a fellow of the American Mathematical Society. == Books == Theory of Linear and Integer Programming (Wiley, 1986, reprinted 1998, ISBN 9780471982326) Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, MR 1261419 Combinatorial Optimization (with William J. Cook, William H. Cunningham, and William R. Pulleyblank, Wiley and Sons, Wiley Series in Discrete Mathematics and Optimization 33, 1998, reprinted 2011, ISBN 9781118031391) Combinatorial Optimization: Polyhedra and Efficiency (Springer, Algorithms and Combinatorics 24, 2003, ISBN 9783540443896) == References ==
Wikipedia:Alexander Shapiro#0	Alexander Shapiro is an A. Russell Chandler III Chair and Professor in H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech. He was editor-in-chief of the journal Mathematical Programming, Series A and was an area editor of the journal Operations Research. Shapiro graduated with M.Sc. degree in mathematics from Moscow State University in 1971 and ten years later got his Ph.D. in applied mathematics and statistics from Ben-Gurion University of the Negev. == References == == External links == Alexander Shapiro publications indexed by Google Scholar
Wikipedia:Alexander Skopin#0	Alexander Ivanovich Skopin (Александр Иванович Скопин) (1927–2003) was a Russian mathematician known for his contributions to abstract algebra. == Biography == Skopin was born on October 22, 1927, in Leningrad, the son of Ivan Alexandrovich Skopin, who was himself also a number theorist and a student of Ivan Matveyevich Vinogradov, and who died in the Siege of Leningrad. After the war, Alexander Skopin studied at Leningrad University, where he was a student of Dmitry Faddeev; From that point to the end of his life, he worked as a researcher at the Steklov Mathematical Institute (where he was scientific secretary from the mid-1960s to the early 1970s) and taught algebra at the St. Petersburg University. He died on September 15, 2003, in St. Petersburg. == Research == Skopin's student work was in abstract algebra, and concerned upper central series of groups and extensions of fields. In the 1970s, Skopin received a second doctorate concerning the application of computer algebra systems to group theory. From that point onward he used computational methods extensively in his research, which focussed on lower central series of Burnside groups. He related this problem to problems in other areas of mathematics including linear algebra and topological sorting of graphs. == References ==
Wikipedia:Alexander V. Karzanov#0	Alexander Viktorovich Karzanov (Russian: Александр Викторович Карзанов, born 1947) is a Russian mathematician known for his work in combinatorial optimization. He is the inventor of preflow-push based algorithms for the maximum flow problem, and the co-inventor of the Hopcroft–Karp–Karzanov algorithm for maximum matching in bipartite graphs. He is a chief researcher at the Federal Research Center "Computer Science and Control" (Institute for System Analysis) of the Russian Academy of Sciences. Karzanov was educated at Moscow State University, completing his doctorate there in 1971. With Georgy Adelson-Velsky and Yefim Dinitz he is the co-author of the book Потоковые алгоритмы [Flow algorithms] (Moscow: Nauka, 1975). He was an invited speaker at the 1990 International Congress of Mathematicians. == References == == External links == Personal home page
Wikipedia:Alexander Varchenko#0	Alexander Nikolaevich Varchenko (Russian: Александр Николаевич Варченко, born February 6, 1949) is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics. == Education and career == From 1964 to 1966 Varchenko studied at the Moscow Kolmogorov boarding school No. 18 for gifted high school students, where Andrey Kolmogorov and Ya. A. Smorodinsky were lecturing mathematics and physics. Varchenko graduated from Moscow State University in 1971. He was a student of Vladimir Arnold. Varchenko defended his Ph.D. thesis Theorems on Topological Equisingularity of Families of Algebraic Sets and Maps in 1974 and Doctor of Science thesis Asymptotics of Integrals and Algebro-Geometric Invariants of Critical Points of Functions in 1982. From 1974 to 1984 he was a research scientist at the Moscow State University, in 1985–1990 a professor at the Gubkin Institute of Gas and Oil, and since 1991 he has been the Ernest Eliel Professor at the University of North Carolina at Chapel Hill. == Research == In 1969 Varchenko identified the monodromy group of a critical point of type A n {\displaystyle A_{n}} of a function of an odd number of variables with the symmetric group S n + 1 {\displaystyle S_{n+1}} which is the Weyl group of the simple Lie algebra of type A n {\displaystyle A_{n}} . In 1971, Varchenko proved that a family of complex quasi-projective algebraic sets with an irreducible base forms a topologically locally trivial bundle over a Zariski open subset of the base. This statement, conjectured by Oscar Zariski, had filled up a gap in the proof of Zariski's theorem on the fundamental group of the complement to a complex algebraic hypersurface published in 1937. In 1973, Varchenko proved René Thom's conjecture that a germ of a generic smooth map is topologically equivalent to a germ of a polynomial map and has a finite dimensional polynomial topological versal deformation, while the non-generic maps form a subset of infinite codimension in the space of all germs. Varchenko was among creators of the theory of Newton polygons in singularity theory, in particular, he gave a formula, relating Newton polygons and asymptotics of the oscillatory integrals associated with a critical point of a function. Using the formula, Varchenko constructed a counterexample to V. I. Arnold's semicontinuity conjecture that the brightness of light at a point on a caustic is not less than the brightness at the neighboring points. Varchenko formulated a conjecture on the semicontinuity of the spectrum of a critical point under deformations of the critical point and proved it for deformations of low weight of quasi-homogeneous singularities. Using the semicontinuity, Varchenko gave an estimate from above for the number of singular points of a projective hypersurface of given degree and dimension. Varchenko introduced the asymptotic mixed Hodge structure on the cohomology, vanishing at a critical point of a function, by studying asymptotics of integrals of holomorphic differential forms over families of vanishing cycles. Such an integral depends on the parameter – the value of the function. The integral has two properties: how fast it tends to zero, when the parameter tends to the critical value, and how the integral changes, when the parameter goes around the critical value. The first property was used to define the Hodge filtration of the asymptotic mixed Hodge structure and the second property was used to define the weight filtration. The second part of the 16th Hilbert problem is to decide if there exists an upper bound for the number of limit cycles in polynomial vector fields of given degree. The infinitesimal 16th Hilbert problem, formulated by V. I. Arnold, is to decide if there exists an upper bound for the number of zeros of an integral of a polynomial differential form over a family of level curves of a polynomial Hamiltonian in terms of the degrees of the coefficients of the differential form and the degree of the Hamiltonian. Varchenko proved the existence of the bound in the infinitesimal 16th Hilbert problem. Vadim Schechtman and Varchenko identified in the Knizhnik–Zamolodchikov equations (or, KZ equations) with a suitable Gauss–Manin connection and constructed multidimensional hypergeometric solutions of the KZ equations. In that construction the solutions were labeled by elements of a suitable homology group. Then the homology group was identified with a multiplicity space of the tensor product of representations of a suitable quantum group and the monodromy representation of the KZ equations was identified with the associated R-matrix representation. This construction gave a geometric proof of the Kohno-Drinfeld theorem on the monodromy of the KZ equations. A similar picture was developed for the quantum KZ equations (or, qKZ-type difference equations) in joint works with Giovanni Felder and Vitaly Tarasov. The weight functions appearing in multidimensional hypergeometric solutions were later identified with stable envelopes in Andrei Okounkov's equivariant enumerative geometry. In the second half of 90s Felder, Pavel Etingof, and Varchenko developed the theory of dynamical quantum groups. Dynamical equations, compatible with the KZ type equations, were introduced in joint papers with G. Felder, Y. Markov, V. Tarasov. In applications, the dynamical equations appear as the quantum differential equations of the cotangent bundles of partial flag varieties. In, Evgeny Mukhin, Tarasov, and Varchenko proved the conjecture of Boris Shapiro and Michael Shapiro in real algebraic geometry: if the Wronski determinant of a complex finite-dimensional vector space of polynomials in one variable has real roots only, then the vector space has a basis of polynomials with real coefficients. It is classically known that the intersection index of the Schubert varieties in the Grassmannian of N-dimensional planes coincides with the dimension of the space of invariants in a suitable tensor product of representations of the general linear group GL N {\displaystyle \operatorname {GL} _{N}} . In, Mukhin, Tarasov, and Varchenko categorified this fact and showed that the Bethe algebra of the Gaudin model on such a space of invariants is isomorphic to the algebra of functions on the intersection of the corresponding Schubert varieties. As an application, they showed that if the Schubert varieties are defined with respect to distinct real osculating flags, then the varieties intersect transversally and all intersection points are real. This property is called the reality of Schubert calculus. == Recognition == Varchenko was an invited speaker at the International Congress of Mathematicians in 1974 in Vancouver (section of algebraic geometry) and in 1990 in Kyoto (a plenary address). In 1973 he received the Moscow Mathematical Society Award. He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to singularity theory, real algebraic geometry, and the theory of quantum integrable systems". He is a Simons Fellow. == Books == Arnolʹd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985. xi+382 pp. ISBN 0-8176-3187-9 Arnolʹd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monographs in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1988. viii+492 pp. ISBN 0-8176-3185-2 Etingof, P.; Varchenko, A. Why the Boundary of a Round Drop Becomes a Curve of Order Four (University Lecture Series), AMS 1992, ISBN 0821870025 Varchenko, A. Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. Advanced Series in Mathematical Physics, 21. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. x+371 pp. ISBN 981-02-1880-X Varchenko, A. Special functions, KZ type equations, and representation theory. CBMS Regional Conference Series in Mathematics, 98. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. viii+118 pp. ISBN 0-8218-2867-3 == References == == External links == Alexander Varchenko at the Mathematics Genealogy Project Varchenko's homepage on the web-site of the University of North Carolina
Wikipedia:Alexander Vasin#0	Alexander Vasin (Russian: Алекса́ндр Алексе́евич Ва́син) (born 1952) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University. Specialist in the field of the theory of non-cooperative games and its applications to economics and biology. He defended the thesis "Evolutionary models and principles of optimality of collective behavior" for the degree of Doctor of Physical and Mathematical Sciences (1991). Was awarded the title of Professor (1994). Author of 6 books and more than 50 scientific articles. == References == == Literature == Evgeny Grigoriev (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 340–341. ISBN 978-5-211-05838-5. == External links == MSU CMC(in Russian) Scientific works of Alexander Vasin Scientific works of Alexander Vasin(in English)
Wikipedia:Alexander Volberg#0	Alexander Volberg (Russian: Александр Львович Вольберг) is a Russian mathematician. He is working in operator theory, complex analysis and harmonic analysis. He received the Salem Prize in 1988 for his work in harmonic analysis. Also he received the Lars Onsager medal in 2004. He is currently a University Distinguished Professor at Michigan State University. From 2007 to 2008 he was the Sir Edmund Whittaker Professor of Mathematical Science at the University of Edinburgh. == Awards and recognition == In 1988, he received the Salem Prize. In 2004, he received the Onsager Medal. In 2011, he won the von Humboldt prize. He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to harmonic analysis and its relations to geometric measure theory". In 2024, the article "On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1", by Fedor Nazarov, Xavier Tolsa, and Alexander Volberg has been selected by the International Congress of Basic Science, as a recipient of the 2024 Frontiers of Science Award in Mathematics. == References == == External links == Alexander Volberg at the Mathematics Genealogy Project Alexander Volberg Homepage Lecture on the Memorial Rubio de Francia Alexander von Humboldt Foundation
Wikipedia:Alexander horned sphere#0	The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander (1924). It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. However, the exterior is not simply connected, unlike the exterior of the usual round sphere. == Construction == The Alexander horned sphere is the particular (topological) embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus: Remove a radial slice of the torus. Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side. Repeat steps 1–2 on the two tori just added ad infinitum. By considering only the points of the tori that are not removed at some stage, an embedding of the sphere with a Cantor set removed results. This embedding extends to a continuous map from the whole sphere, which is injective (hence a topological embedding since the sphere is compact) since points in the sphere approaching two different points of the Cantor set will end up in different 'horns' at some stage and therefore have different images. == Impact on theory == The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. The exterior is not simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. This shows that the Jordan–Schönflies theorem does not hold in three dimensions, as Alexander had originally thought. Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between the categories of topological manifolds, differentiable manifolds, and piecewise linear manifolds became apparent. Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere. Although the solid horned sphere is not a manifold, R. H. Bing showed that its double (which is the 3-manifold obtained by gluing two copies of the horned sphere together along the corresponding points of their boundaries) is in fact the 3-sphere. One can consider other gluings of the solid horned sphere to a copy of itself, arising from different homeomorphisms of the boundary sphere to itself. This has also been shown to be the 3-sphere. The solid Alexander horned sphere is an example of a crumpled cube; i.e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere. == Generalizations == One can generalize Alexander's construction to generate other horned spheres by increasing the number of horns at each stage of Alexander's construction or considering the analogous construction in higher dimensions. Other substantially different constructions exist for constructing such "wild" spheres. Another example, also found by Alexander, is Antoine's horned sphere, which is based on Antoine's necklace, a pathological embedding of the Cantor set into the 3-sphere. == See also == Cantor tree surface Wild knot Wild arc, specifically the Fox–Artin arc == References == === Citations === Alexander, J. W. (1924), "An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected", Proceedings of the National Academy of Sciences of the United States of America, 10 (1), National Academy of Sciences: 8–10, Bibcode:1924PNAS...10....8A, doi:10.1073/pnas.10.1.8, ISSN 0027-8424, JSTOR 84202, PMC 1085500, PMID 16576780 Fuchs, Dmitry; Tabachnikov, Serge (2007), Mathematical Omnibus. 30 Lectures on Classical Mathematics, Providence, RI: American Mathematical Society, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979 Hatcher, Allen, Algebraic Topology, http://pi.math.cornell.edu/~hatcher/AT/ATpage.html Hocking, John Gilbert; Young, Gail Sellers (1988) [1961]. Topology. Dover. ISBN 0-486-65676-4. Spivak, Michael (1999). A comprehensive introduction to differential geometry (Volume 1). Publish or Perish. ISBN 0-914098-70-5. == External links == Weisstein, Eric W. "Alexander's Horned Sphere". MathWorld. Zbigniew Fiedorowicz. Math 655 – Introduction to Topology. [1] Archived 2005-08-25 at the Wayback Machine – Lecture notes Construction of the Alexander sphere rotating animation PC OpenGL demo rendering and expanding the cusp
Wikipedia:Alexandra Bellow#0	Alexandra Bellow (née Bagdasar; previously Ionescu Tulcea; August 30, 1935 – May 2, 2025) was a Romanian-American mathematician, who made contributions to the fields of ergodic theory, probability and analysis. == Biography == Bellow was born in Bucharest, Romania, on August 30, 1935, as Alexandra Bagdasar. Her parents were both physicians. Her mother, Florica Bagdasar (née Ciumetti), was a child psychiatrist. Her father, Dumitru Bagdasar, was a neurosurgeon. She received her M.S. in mathematics from the University of Bucharest in 1957, where she met and married her first husband, mathematician Cassius Ionescu-Tulcea. She accompanied her husband to the United States in 1957 and received her Ph.D. from Yale University in 1959 under the direction of Shizuo Kakutani with thesis Ergodic Theory of Random Series. After receiving her degree, she worked as a research associate at Yale from 1959 until 1961, and as an assistant professor at the University of Pennsylvania from 1962 to 1964. From 1964 until 1967, she was an associate professor at the University of Illinois at Urbana–Champaign. In 1967, she moved to Northwestern University as a Professor of Mathematics. She was at Northwestern until her retirement in 1996, when she became Professor Emeritus. During her marriage to Cassius Ionescu-Tulcea (1956–1969), she and her husband co-wrote many papers and a research monograph on lifting theory. Alexandra's second husband was the writer Saul Bellow, who was awarded the Nobel Prize in Literature in 1976, during their marriage (1975–1985). Alexandra features in Bellow's writings; she is portrayed lovingly in his memoir To Jerusalem and Back (1976), and, his novel The Dean's December (1982), more critically, satirically in his last novel, Ravelstein (2000), which was written many years after their divorce. The decade of the nineties was for Alexandra a period of personal and professional fulfillment, brought about by her marriage in 1989 to the mathematician Alberto P. Calderón. Bellow died on May 2, 2025, at the age of 89. == Mathematical work == Some of her early work involved properties and consequences of lifting. Lifting theory, which had started with the pioneering papers of John von Neumann and later Dorothy Maharam, came into its own in the 1960s and 1970s with the work of the Ionescu Tulceas and provided the definitive treatment for the representation theory of linear operators arising in probability, the process of disintegration of measures. Their Ergebnisse monograph from 1969 became a standard reference in this area. By applying a lifting to a stochastic process, the Ionescu Tulceas obtained a ‘separable’ process; this gives a rapid proof of Joseph Leo Doob's theorem concerning the existence of a separable modification of a stochastic process (also a ‘canonical’ way of obtaining the separable modification). Furthermore, by applying a lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly measurable function; this gives a one line proof of Phillips's classical theorem (also a ‘canonical’ way of obtaining the strongly measurable version). We say that a set H of measurable functions satisfies the "separation property" if any two distinct functions in H belong to distinct equivalence classes. The range of a lifting is always a set of measurable functions with the "separation property". The following ‘metrization criterion’ gives some idea why the functions in the range of a lifting are so much better behaved. Let H be a set of measurable functions with the following properties: (I) H is compact (for the topology of pointwise convergence); (II) H is convex; (III) H satisfies the "separation property". Then H is metrizable. The proof of the existence of a lifting commuting with the left translations of an arbitrary locally compact group, by the Ionescu Tulceas, is highly non-trivial; it makes use of approximation by Lie groups, and martingale-type arguments tailored to the group structure. In the early 1960s, she worked with C. Ionescu Tulcea on martingales taking values in a Banach space. In a certain sense, this work launched the study of vector-valued martingales, with the first proof of the ‘strong’ almost everywhere convergence for martingales taking values in a Banach space with (what later became known as) the Radon–Nikodym property; this, by the way, opened the doors to a new area of analysis, the "geometry of Banach spaces". These ideas were later extended by Bellow to the theory of ‘uniform amarts’, (in the context of Banach spaces, uniform amarts are the natural generalization of martingales, quasi-martingales and possess remarkable stability properties, such as optional sampling), now an important chapter in probability theory. In 1960, Donald Samuel Ornstein constructed an example of a non-singular transformation on the Lebesgue space of the unit interval, which does not admit a σ {\displaystyle \sigma } –finite invariant measure equivalent to Lebesgue measure, thus solving a long-standing problem in ergodic theory. A few years later, Rafael V. Chacón gave an example of a positive (linear) isometry of L 1 {\displaystyle L_{1}} for which the individual ergodic theorem fails in L 1 {\displaystyle L_{1}} . Her work unifies and extends these two remarkable results. It shows, by methods of Baire category, that the seemingly isolated examples of non-singular transformations first discovered by Ornstein and later by Chacón, were in fact the typical case. Beginning in the early 1980s, Bellow began a series of papers that brought about a revival of that area of ergodic theory dealing with limit theorems and the delicate question of pointwise a.e. convergence. This was accomplished by exploiting the interplay with probability and harmonic analysis, in the modern context (the Central limit theorem, transference principles, square functions and other singular integral techniques are now part of the daily arsenal of people working in this area of ergodic theory) and by attracting a number of talented mathematicians who were very active in this area. One of the two problems that she raised at the Oberwolfach meeting on "Measure Theory" in 1981, was the question of the validity, for f {\displaystyle f} in L 1 {\displaystyle L_{1}} , of the pointwise ergodic theorem along the ‘sequence of squares’, and along the ‘sequence of primes’ (A similar question was raised independently, a year later, by Hillel Furstenberg). This problem was solved several years later by Jean Bourgain, for f {\displaystyle f} in L p {\displaystyle L_{p}} , p > 1 {\displaystyle p>1} in the case of the "squares", and for p > ( 1 + 3 ) / 2 {\displaystyle p>(1+{\sqrt {3}})/2} in the case of the "primes" (the argument was pushed through to p > 1 {\displaystyle p>1} by Máté Wierdl; the case of L 1 {\displaystyle L_{1}} however has remained open). Bourgain was awarded the Fields Medal in 1994, in part for this work in ergodic theory. It was Ulrich Krengel who first gave, in 1971, an ingenious construction of an increasing sequence of positive integers along which the pointwise ergodic theorem fails in L 1 {\displaystyle L_{1}} for every ergodic transformation. The existence of such a "bad universal sequence" came as a surprise. Bellow showed that every lacunary sequence of integers is in fact a "bad universal sequence" in L 1 {\displaystyle L_{1}} . Thus lacunary sequences are ‘canonical’ examples of "bad universal sequences". Later she was able to show that from the point of view of the pointwise ergodic theorem, a sequence of positive integers may be "good universal" in L p {\displaystyle L_{p}} , but "bad universal" in L q {\displaystyle L_{q}} , for all 1 ≤ q < p {\displaystyle 1\leq q<p} . This was rather surprising and answered a question raised by Roger Jones. A place in this area of research is occupied by the "strong sweeping out property" (that a sequence of linear operators may exhibit). This describes the situation when almost everywhere convergence breaks down even in L ∞ {\displaystyle L_{\infty }} and in the worst possible way. Instances of this appear in several of her papers. The "strong sweeping out property" plays an important role in this area of research. Bellow and her collaborators did an extensive and systematic study of this notion, giving various criteria and numerous examples of the strong sweeping out property. Working with Krengel, she was able to give a negative answer to a long-standing conjecture of Eberhard Hopf. Later, Bellow and Krengel working with Calderón were able to show that in fact the Hopf operators have the "strong sweeping out" property. In the study of aperiodic flows, sampling at nearly periodic times, as for example, t n = n + ε ( n ) {\displaystyle t_{n}=n+\varepsilon (n)} , where ε {\displaystyle \varepsilon } is positive and tends to zero, does not lead to a.e. convergence; in fact strong sweeping out occurs. This shows the possibility of serious errors when using the ergodic theorem for the study of physical systems. Such results can be of practical value for statisticians and other scientists. In the study of discrete ergodic systems, which can be observed only over certain blocks of time, one has the following dichotomy of behavior of the corresponding averages: either the averages converge a.e. for all functions in L 1 {\displaystyle L_{1}} , or the strong sweeping out property holds. This depends on the geometric properties of the blocks. Several mathematicians (including Bourgain) worked on problems posed by Bellow and answered those questions in their papers. == Academic honors, awards, recognition == 1977–80 Member, Visiting Committee, Harvard University Mathematics Department 1980 Fairchild Distinguished Scholar Award, California Institute of Technology, Winter Term 1987 Humboldt Prize, Alexander von Humboldt Foundation, Bonn, Germany 1991 Emmy Noether Lecture, San Francisco 1997 International Conference in Honor of Alexandra Bellow, on the occasion of her retirement, held at Northwestern University, October 23–26, 1997. A Proceedings of this Conference appeared as a special issue of the Illinois Journal of Mathematics, Fall 1999, Vol. 43, No. 3. 2017 class of Fellows of the American Mathematical Society "for contributions to analysis, particularly ergodic theory and measure theory, and for exposition". == Professional editorial activities == 1974–77 Editor, Transactions of the American Mathematical Society 1980–82 Associate Editor, Annals of Probability 1979– Associate Editor, Advances in Mathematics == See also == Saul Bellow == Selected publications == == References ==
Wikipedia:Alexandra Seceleanu#0	Alexandra Seceleanu is a Romanian mathematician specializing in commutative algebra. She is an associate professor of mathematics at the University of Nebraska–Lincoln. She was awarded the 2024–2025 Ruth I. Michler Memorial Prize. == Education and career == Seceleanu graduated from the University of Bucharest, and in 2005 obtained a master's degree in mathematics from the Bucharest Superior Normal School with a focus on algebra, geometry, and topology. She completed a Ph.D. at the University of Illinois in 2011, supervised by Hal Schenck. Seceleanu started working as a postdoctoral researcher at the University of Nebraska–Lincoln in 2011, where in 2015 she became an assistant professor, and in 2021 an associate professor. She is on the editorial board of the Journal of Commutative Algebra. === Research === Seceleanu's research within the field of commutative algebra concerns both theoretical and computational problems. She has an interest in algebraic geometry, in particular in how it can be studied via methods from homological algebra. She also works on computational algebra, and has contributed to Macaulay2. == Recognition == In 2018 Seceleanu won the Harold & Esther Edgerton Junior Faculty Award, which is awarded annually by the University of Nebraska–Lincoln to a junior faculty member demonstrating "creative research, extraordinary teaching abilities, and academic promise". She was awarded the 2024-2025 Ruth I. Michler Memorial Prize. == References == == External links == Home page Alexandra Seceleanu publications indexed by Google Scholar
Wikipedia:Alexandre Eremenko#0	Alexandre Emanuilovych Eremenko (born 1954) is a Ukrainian-American mathematician who works in the fields of complex analysis and dynamical systems. == Academic career == Eremenko was born into a medical family in Kharkiv, Ukraine. His father was a pathophysiologist, professor and head of the Department of pathophysiology at Ternopil National Medical University. His mother was an ophthalmologist. He obtained his master's degree from Lviv University in 1976 and worked in the Institute of Low temperature physics and Engineering in Kharkiv until 1990. He received his PhD from Rostov State University in 1979 (Asymptotic Properties of Meromorphic and Subharmonic Functions), and is currently a distinguished professor at Purdue University. In complex dynamics, Eremenko explored escaping sets at the iteration of entire, transcendent functions and conjectured that the connected components of this escaping set are unbounded (Eremenko's conjecture). The conjecture is still open, and has been called "one of the most famous open problems in transcendental dynamics". == Distinctions == Eremenko was a recipient of the Humboldt Prize in Mathematics. In 2013, he became a fellow of the American Mathematical Society, for "contributions to value distribution theory, geometric function theory, and other areas of analysis and complex dynamics". He was an invited speaker in the International congress of mathematicians in Beijing in 2002. == See also == Escaping set Polynomial lemniscate Gravity train Paul Eremenko == Notes == == References == == External links == Alexandre Eremenko at the Mathematics Genealogy Project
Wikipedia:Alexandre Proutiere#0	Alexandre Proutiere is a professor of Electrical Engineering at KTH, the Royal Institute of Technology. He received an engineering degree from Ecole Nationale Superieure des Telecoms (Paris) and then, from 1998 to 2000, he worked in the radio communication department at the Ministry of Foreign Affairs in Paris. He received his PhD in Applied Mathematics from Ecole Polytechnique, Palaiseau, France in 2003 under the supervision of James Roberts. Following his PhD he worked as a researcher at Microsoft Research in Cambridge (UK) before joining KTH as an associate professor. == Research == Proutiere's research focuses on the design and performance evaluation of computer networks, with a specific interest in resource allocation and control in wireless systems. His work has had significant impact both in the development of theoretical tools and in their practical application. This is highlighted by the "Best Paper" awards he has received from top publication venues such as ACM SIGMETRICS and ACM Mobihoc. Additionally, his work has been recognized by the ACM SIGMETRICS Rising Star award, given for outstanding contributions to computer/communication performance evaluation by a research not more than seven years from their PhD. == References ==
Wikipedia:Alexandru Ghika#0	Alexandru Ghika (June 22, 1902 – April 11, 1964) was a Romanian mathematician, founder of the Romanian school of functional analysis. == Life == He was born in Bucharest, into the Ghica family, the son of Ioan Ghika (1873–1949) and Elena Metaxa (1870–1951), and great-great-grandson of Grigore IV Ghica, Prince of Wallachia. He started his secondary studies at the Gheorghe Lazăr High School in Bucharest. In 1917, he left with his family for Paris, completing his secondary studies at the Lycée Louis-le-Grand in 1920. He then entered the University of Paris (the Sorbonne) with a major in mathematics, graduating in 1922. In 1929, he obtained a Ph.D. in mathematics from the Faculté des Sciences of the University of Paris, with thesis written under the direction of Arnaud Denjoy. After completing his doctorate, Ghika returned to Romania. In November 1932 he became assistant professor in the Mathematics Department of the University of Bucharest, working in the Function Theory section chaired by Dimitrie Pompeiu. On February 7, 1935, he was promoted to associate professor, and in 1945 he was named Full Professor and chair of the newly founded Functional Analysis section. In 1935, Ghika was elected corresponding member of the Romania Academy of Sciences, being promoted to full member in 1938. In 1955 he became corresponding member of the Romanian Academy, and was promoted to full membership on March 20, 1963. In 1949, at the founding of the Institute of Mathematics of the Romanian Academy, he became the chair of the Functional Analysis section of that Institute, a position he held till his death. Ghika married Elisabeta Angelescu (daughter of one-time Prime Minister Constantin Angelescu) on June 7, 1934. They had a son, Grigore (born November 7, 1936), who became a researcher at the Institute of Atomic Physics in Măgurele. Alexandru Ghika died in Bucharest of lung cancer. He was buried at the Teiul Doamnei Ghica Church, founded in 1833 by Prince Grigore Alexandru Ghica. In March, 2007, the heirs of the Ghika and Angelescu families won back their rights to Hotel Lido, in Bucharest. == Legacy == Ghika introduced the study of functional analysis in Romania at a time when it was still a new field of mathematics. As a professor at the University of Bucharest, he was an important promoter of the Bourbaki rigorous approach to mathematics. His collected works number 103 published articles and books, 10 manuscripts, and 25 didactic works. His research in functional analysis introduced new concepts, such as F-ordered rings (now known as Ghika rings), which have the property that in any module over a ring in this class, the analogue of the Hahn–Banach theorem holds. He is also remembered for a representation theorem for reflexive Banach spaces, now known as the Ghika–James representation theorem. Among his doctoral students were Silviu Teleman and Petru Mocanu. == Works == Sur les fonctions de carré sommable le long des contours de leur domaines d’holomorphisme et leur applications aux équations différentielles linéaires d’ordre infini, Ph.D. thesis, Faculté des Sciences de Paris, 1929. OCLC 42822939 On reflexive Banach spaces, Acad. Repub. Pop. Române. Bul. Ști. A. 1 (1949), 639–644. MR0038015 The extension of general linear functionals in semi-normed modules, Acad. Repub. Pop. Române Bul. Ști. Ser. Mat. Fiz. Chim. 2 (1950), 399–405. MR0045319 Analiză funcțională (Functional analysis), Editura Academiei Republicii Socialiste România, Bucharest, 1967. MR0211242 Opera matematică (Mathematical works), Editura Academiei Republicii Socialiste România, Bucharest, 1968. MR0229502 == References == == External links == (in Romanian) Short biography
Wikipedia:Alexei Gvishiani#0	Alexei Dzhermenovich Gvishiani (Russian: Алексей Джерменович Гвишиани; 29 October 1948) is a well-known Russian scientist, full member (academician) of the Russian Academy of Sciences (RAS). Chief scientist of the Geophysical Center of RAS. Member of the Scientific Coordinating Council of the Federal Agency of Scientific organizations of Russia (FASO) and the Expert Council of the Russian Scientific Foundation. Foreign member of the Romanian Academy of Engineering and Technical Sciences and the National Academy of Sciences of Ukraine. Doctor Honoris Causa of the National Technological University of Ukraine, professor of Lomonosov Moscow State University and the Paris Institute of Earth Physics of the Earth. Chair of the Russian Geophysical Committee and CODATA Committee of the Russian Academy of Sciences, vice-president of CODATA in 2002-2006. Vice-chair of the WDC Panel of the International Council for Science (ICSU) (1996–2006). Deputy chairman of the Committee of System Analysis, RAS. Vice-president of the Scientific Council International Institute for Applied Systems Analysis (IIASA) and IIASA Program Committee chair (2010–2014). Member of Academia Europaea (2017). == Biography and academic career == Gvishiani was born on October 29, 1948, in Moscow in the family of a prominent philosopher, expert in control theory and systems analysis, full member (academician) of the USSR Academy of Sciences, Dzhermen Gvishiani. Through Dzhermen he is the grandson of Mikhail Gvishiani, the NKVD general. On his mother's side Gvishiani is a grandson of Alexei Kosygin, Chairman of the USSR Council of Ministers in 1964–1980. After graduating from Lomonosov Moscow State University (MSU), А. Gvishiani entered for postgraduate education at the chair of the theory of functions and functional analysis, Department of mathematics and mechanics of MSU. He was a student of an internationally famous mathematician Prof. Izrael M. Gelfand. During Gvishiani's postgraduate studies he published a number of papers in leading mathematical journals. His main result was the proof of the Gelfand-Graev-Gvishiani formula for the characters of discrete series representations for the group of matrices with elements from a local non-Archimedean field (1973–74). During his university studies he was influenced by prominent Soviet scientists – I. Gelfand, М. Keldysh, V. Chelomey, V. Sadovnichy and V. Maslov (presently academicians), M. Graev, А. Kirillov. In 1974 Gvishiani defended his PhD thesis devoted to the theory of functions and functional analysis. From 1974 to 1978 Gvishiani worked as assistant and then as associate professor at the Department of mathematics and mechanics of MSU. In 1978 Gvishiani took a research position in the Schmidt Institute of Physics of the Earth of the Academy of Sciences. His career in this Institute developed from a research assistant starting as a research assistant up to deputy director general. In 1983, Gvishiani defended his doctoral thesis in geophysics at the Schmidt Institute of Physics of the Earth. In 1991–2005 Gvishiani established and headed the Center for geophysical data studies and network technologies of IPE RAS. During his work at the Center he created theoretical foundations of geoinformatics as a part of mathematical geophysics. In 2005 he was elected the director of the Geophysical Center of the Russian Academy of Sciences (GC RAS) and remained in that position till 2018. In 2018 by the decree of the Bureau of the RAS Earth Sciences Division he was appointed the chief scientist of GC RAS. In 2006 he was elected a corresponding member of the Russian Academy of Sciences in geoinformatics, and in 2011 – a full member of Russian Academy of Sciences in geology and geo-informatics. Since 2007 he is a member of the bureau, and since 2008 Deputy Academician Secretary of the Earth Sciences Division of RAS. He is also a member of the RAS Scientific Council on the complex problems of Eurasian economic integration, modernization, competitiveness and sustainable development and Fellow of the Russian Geographical Society. == Scientific results achieved == New features of non-Archimedean Radon transformation and the following study of representations of the group G=SL(2) in a functional space over Lobachevsky plane (1974–1978); Introduction of a new family of classification algorithms "Voting by the Set of Feature (VFS)" for studying dynamic recognition problems (1979–1985); New dichotomy classification algorithms with single learning class (since 2016) Pattern recognition system FCAZ (Fuzzy Clustering and Zoning) to determine areas of the highest seismic risk (2014–2015) Construction of a mathematical model of recognition of possible strong earthquakes epicenters locations and their time stability as a basis for seismic zoning; recognition of possible strongest, strong and significant earthquake occurrence locations in seismic regions of the Earth and evaluation of their reliability using the models for dynamic pattern recognition (1978–1990 and 2010–2013); Hierarchic clustering of spatial geophysical and geodynamic objects using discrete mathematical analysis; original algorithms of time series anomaly recognition on the basis of the fuzzy sets theory and their application to volcano monitoring and to the analysis of gravity and geomagnetic data (1999–2012); Development and integration into a single network of a system of magnetic observatories in the Russian Federation, (the highest standard of quality INTERMAGNET), integrated by a common center for geomagnetic data acquisition and analysis (2009–2013); Unique original automated hardware-software system to monitor and recognize geomagnetic activity using the network of observatories and satellite data (2014–2016); System analysis and in magnetic and other geophysical and geological data studies. New methods of magnetic activity recognition using observatory and satellite data (since 2006). == International scientific research == Gvishiani made an outstanding contribution to the development of cooperation between Russian Academy of Sciences and the International Institute for Applied Systems Analysis (IIASA)/ as well as to promotion of cooperation between Russian scientists and the international scientific community. Academician Gvishiani has been active in the field of international scientific cooperation: Vice President of the European Mediterranean seismological center (EMSC/CSEM) (EMSC/CSEM) (1996–2003); principal researcher of the joint project of RAS, RFBR and Scientific Research Center of France (CNRS) "Identification of areas of earthquake possible occurrence in regions of moderate seismicity" (PICS) (1980–1986, 1998–2001) and "Monitoring of basalt volcanoes with the use of artificial intelligence methods" (PICS) (2000–2006); delegate of RAS in the International Union of Geodesy and Geophysics (IUGG) (2007–2012) and CODATA (since 2005). == Teaching and lecturing == In 1992–2000 Gvishiani was professor at the Chair of the theory of functions and functional analysis, MSU Department of mathematics and mechanics. He was a supervisor of two doctoral and twelve PhD theses. А. Gvishiani was a visiting researcher in many scientific organizations in different countries, including the Institute of Physics of the Earth in Paris (Institut de Physique du Globe de Paris), Institute of Geography of Spain, American and German universities and research institutes. == Editorial and publishing activity == He is the author of five books published in Russian, English, French, Hungarian, and Italian languages. More than 300 research papers were published in international and Russian scientific peer-reviewed journals. Works by A.D. Gvishiani were co-authored by Russian academicians I.M. Gelfand, M.A. Sadovsky, V.N. Strakhov, V.I. Keilis-Borok, I.F. Obraztsov, V.I. Osipov and French academicians Jean-Louis LeMouél and Jean Coulomb. Over the years Gvishiani actively participates in the work of the International Council for Science (ICSU), holding various positions: Chairman of the coordinating Committee “Data Centers and Data Exchange” and Bureau member of the International Program “Lithosphere” (1985–1995); head of a number of CODATA WGs; member of the ICSU Strategic Committee on Information and Data (SCID) (2007–2008); Liaison officer of IUGG in CODATA (ICSU) (2011–2019). == Awards and honours == Order of Friendship (2009) Medal "In Commemoration of the 850th Anniversary of Moscow"(1997) Medal "50 Years of the Mongolian People's Revolution" (1971) == Family == Wife – Natalia B. Gvishiani – Professor of the Philological Department of Lomonosov Moscow State University; daughter Ekaterina A. Semenikhina - economist-mathematician, Head of "Ekaterina" Fund, Russian Honorary Consul General in Monaco. == References == Zoback, Mary Lou; Zoback, Mark D.; Adams, J.; Assumpção, M.; Bell, S.; Bergman, E. A.; Blümling, P.; Brereton, N. R.; Denham, D.; Ding, J.; Fuchs, K.; Gay, N.; Gregersen, S.; Gupta, H. K.; Gvishiani, A.; Jacob, K.; Klein, R.; Knoll, P.; Magee, M.; Mercier, J. L.; Müller, B. C.; Paquin, C.; Rajendran, K.; Stephansson, O.; Suarez, G.; Suter, M.; Udias, A.; Xu, Z. H.; Zhizhin, M. (28 September 1989). "Global patterns of tectonic stress". Nature. 341 (6240): 291–298. Bibcode:1989Natur.341..291Z. doi:10.1038/341291a0. S2CID 4284009. ZLOTNICKI, J; LEMOUEL, J; GVISHIANI, A; AGAYAN, S; MIKHAILOV, V; BOGOUTDINOV, S; KANWAR, R; YVETOT, P (30 May 2005). "Automatic fuzzy-logic recognition of anomalous activity on long geophysical records: Application to electric signals associated with the volcanic activity of La Fournaise volcano (Réunion Island)". Earth and Planetary Science Letters. 234 (1–2): 261–278. doi:10.1016/j.epsl.2005.01.040. Widiwijayanti, Christina; Mikhailov, Valentin; Diament, Michel; Deplus, Christine; Louat, Rémy; Tikhotsky, Sergei; Gvishiani, Alexei (October 2003). "Structure and evolution of the Molucca Sea area: constraints based on interpretation of a combined sea-surface and satellite gravity dataset". Earth and Planetary Science Letters. 215 (1–2): 135–150. Bibcode:2003E&PSL.215..135W. doi:10.1016/S0012-821X(03)00416-3. Gvishiani, Alexei; Lukianova, Renata; Soloviev, Anatoly; Khokhlov, Andrei (16 July 2014). "Survey of Geomagnetic Observations Made in the Northern Sector of Russia and New Methods for Analysing Them". Surveys in Geophysics. 35 (5): 1123–1154. Bibcode:2014SGeo...35.1123G. doi:10.1007/s10712-014-9297-8. S2CID 128623364. Soloviev, A. A.; Agayan, S. M.; Gvishiani, A. D.; Bogoutdinov, Sh. R.; Chulliat, A. (15 May 2012). "Recognition of disturbances with specified morphology in time series: Part 2. Spikes on 1-s magnetograms". Izvestiya, Physics of the Solid Earth. 48 (5): 395–409. Bibcode:2012IzPSE..48..395S. doi:10.1134/S106935131204009X. S2CID 129578408. Gvishiani, A. D.; Dzeboev, B. A.; Agayan, S. M. (7 November 2013). "A new approach to recognition of the strong earthquake-prone areas in the Caucasus". Izvestiya, Physics of the Solid Earth. 49 (6): 747–766. Bibcode:2013IzPSE..49..747G. doi:10.1134/S1069351313060049. S2CID 55491070. Dubois, Alexej Gvishiani, Jacques O. (2010). Artificial intelligence and dynamic systems for geophysical applications. Berlin: Springer. ISBN 978-3-642-07757-9.{{cite book}}: CS1 maint: multiple names: authors list (link) Gvishiani, J.O. Dubois, A. (1998). Dynamic systems and dynamic classification problems in geophysical applications. Berlin: Springer. p. 256. ISBN 978-3-642-49951-7.{{cite book}}: CS1 maint: multiple names: authors list (link) Kirillov, A. A.; Gvishiani, A. D. (1982). Theorems and problems in functional analysis. Series "Problem Books in Mathematics". U.S.A.: Springer-Verlag New York. p. 347. ISBN 978-1-4613-8155-6. Gvishiani, A.D.; et al. (1988). Pisarenko, V.F. (ed.). Prognozirovanie mest zemletri︠a︡seniĭ v regionakh umerennoĭ seĭsmichnosti (Nauch. izd. ed.). Moskva. p. 175. ISBN 502000720X. {{cite book}}: \|work= ignored (help)CS1 maint: location missing publisher (link) Gurvich, A.D. Grishiani, V.A. (1992). Dinamicheskie zadachi klassifikat︠s︡ii i vypukloe programmirovanie v prilozhenii︠a︡kh. Moskva: "Nauka," GLav. red. fiziko-matematicheskoĭ lit-ry. p. 355. ISBN 5020145815.{{cite book}}: CS1 maint: multiple names: authors list (link) Gvishiani, Aleksei (2004). Fenomen Kosygina : zapiski vnuka. Mnenija sovremennikov. Moskva: Fond Kul'tury "Ekaterina". p. 308. ISBN 5-86863-191-9.
Wikipedia:Alexei Kostrikin#0	Alexei Ivanovich Kostrikin (Russian: Алексей Иванович Кострикин) (12 February 1929 – 22 September 2000) was a Russian mathematician, specializing in algebra and algebraic geometry. == Life == Kostrikin graduated from the Faculty of Mechanics and Mathematics at Moscow State University in 1952. In 1960, he earned a Doctor of Sciences degree under Igor Shafarevich at the Steklov Institute of Mathematics with a thesis on the Burnside problem. He became a faculty member at Moscow State University in 1963. == Prizes == Kostrikin was awarded the USSR State Prize in 1968 for his research on finite groups and Lie algebras and was elected the corresponding member of the USSR Academy of Sciences in 1976. == Books == Alexei Kostrikin published many scientific articles, books and textbooks, including a university textbook about algebra Introduction to algebra, translated into English and other languages. == Selected publications == Around Burnside, Springer Verlag 1990 2012 pbk reprint with Pham Huu Tiep: Orthogonal Decompositions and Integral Lattices, de Gruyter 1994 with Yuri Manin: Linear algebra and geometry, Gordon and Breach 1989; 1997 pbk edition Introduction to Algebra, Springer Verlag 1982 (Russian original 1977) Exercises in algebra: a collection of exercises in algebra, linear algebra and geometry, Gordon and Breach 1996 == Further reading == V. A. Artamonov, Yu. A .Bahturin, I. A. Chubarov, et al., Dedication: Alexei Ivanovich Kostrikin, Comm. Algebra 29 (9) (2001), ix–xiv. Preface: On the 70th birthday of Alexei Ivanovich Kostrikin, J. Math. Sci. (New York) 93 (6) (1999), 801–808. E. B. Vinberg, E. S. Golod, E. I. Zelmanov, et al., Aleksei Ivanovich Kostrikin [1929–2000] (Russian), Uspekhi Mat. Nauk 56 3(339) (2001), 143–145. E. B. Vinberg, E. S. Golod, E. I. Zelmanov, et al., Aleksei Ivanovich Kostrikin [1929–2000], Russian Math. Surveys 56 (3) (2001), 559–561. == References ==
Wikipedia:Alexei Venkov#0	Alexei Borisovich Venkov (Алексей Борисович Венков, born 1946) is a Russian mathematician, specializing in the spectral theory of automorphic forms. Venkov graduated from Leningrad State University in 1969 and received there in 1973 his Russian candidate degree (Ph.D.) under Ludvig Faddeev. He then became an academic at the Steklov Institute in Saint Petersburg, where he received in 1980 his Russian doctorate (higher doctoral degree) with dissertation Spectral theory of automorphic functions (Russian). He was a visiting scholar at IHES, at the University of Göttingen, in Paris (University of Paris VI, École Normale Superieure, Institute Henri Poincaré), at the MSRI, at Stanford University, several times at the Max Planck Institute for Mathematics in Bonn, at the University of Lille, and at the Aarhus University. Since 2001 he has been a lecturer at Aarhus University. Venkov's research deals with the spectral theory of automorphic forms and their applications in number theory and mathematical physics. He has proved partial results for the Roelcke-Selberg conjecture. In 1983 he was an Invited Speaker at the ICM in Warsaw. In 2006 he received the Humboldt Research Award. == Selected publications == === Articles === with V. L. Kalinin and Ludvig Faddeev: A nonarithmetic derivation of the Selberg trace formula, Journal of Soviet Mathematics, vol. 8, 1977, pp. 171–199 Spectral theory of automorphic functions, the Selberg zeta-function, and some problems of analytic number theory and mathematical physics, Russian Mathematical Surveys, vol. 34, 1979, pp. 79–153 Remainder term in the Weyl-Selberg asymptotic formula, Journal of Mathematical Sciences 17, no. 5, 1981, pp. 2083–2097 doi:10.1007/BF01567587 with N. V. Proskurin: Automorphic forms and Kummer's problem, Russian Mathematical Surveys, vol. 37, 1982, pp. 165–190 Selberg's trace formula for an automorphic Schroedinger Operator, Functional Analysis and Applications, vol. 25, 1991, pp. 102–111 doi:10.1007/BF01079589 On a multidimensional variant of the Roelcke-Selberg conjecture, Saint Petersburg Mathematical Journal, vol. 4, 1993, pp. 527–538 with A. M. Nikitin: The Selberg trace formula, Ramanujan graphs and some problems in mathematical physics, Saint Petersburg Mathematical Journal, vol. 5, 1994, pp. 419–484. Approximation of Maass forms by analytic modular forms, Saint Petersburg Mathematical Journal, vol. 6, 1995, pp. 1167–1177 The Zagier formula with the Eisenstein-Maass series at odd integer points, and the generalized Selberg zeta function, Saint Petersburg Mathematical Journal, vol. 6, 1995, pp. 519–527. with E. Balslev: Selberg's eigenvalue conjecture and the Siegel zeros for Hecke L-series, in: Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto 1997, Advanced Studies in Pure Mathematics 26, Mathematical Society of Japan 2000, pp. 19–32 with Erik Balslev: Spectral theory of Laplacians for Hecke groups with primitive character, Acta Mathematica, vol. 186, 2001, pp. 155–217, doi:10.1007/BF02401839; Correction vol. 192, 2004, pp. 1–3 doi:10.1007/BF02441083 with E. Balslev: On the relative distribution of eigenvalues of exceptional Hecke operators and automorphic Laplacians, Original publication: Algebra i Analiz, tom 17 (2005), nomer 1. Journal: St. Petersburg Math. J. 17 (2006), 1-37 doi:10.1090/S1061-0022-06-00891-0 with A. Momeni: Mayer's transfer operator approach to Selberg's zeta function, Original publication: Algebra i Analiz, tom 24 (2012), nomer 4. Journal: St. Petersburg Math. J. 24 (2013), 529–553 doi:10.1090/S1061-0022-2013-01252-0 with D. Mayer and A. Momeni: Congruence properties of induced representations and their applications, Original publication: Algebra i Analiz, tom 26 (2014), nomer 4. Journal: St. Petersburg Math. J. 26 (2015), 593–606 doi:10.1090/spmj/1352 === Books === Spectral theory of automorphic functions, American Mathematical Society 1983 Spectral theory of automorphic functions and its applications, Kluwer 1990; 2012 reprint. Springer. 6 December 2012. ISBN 9789400918924; pbk{{cite book}}: CS1 maint: postscript (link) == References == == External links == Alexei Venkov, Centre for Quantum Geometry for Moduli Spaces, Aarhus University Archived 2018-03-10 at the Wayback Machine Venkov Aleksei Borisovich, mathnet.ru
Wikipedia:Alexey Ivakhnenko#0	Alexey Ivakhnenko (Ukrainian: Олексíй Григо́рович Іва́хненко; 30 March 1913 – 16 October 2007) was a Soviet and Ukrainian mathematician most famous for developing the group method of data handling (GMDH), a method of inductive statistical learning, for which he is considered as one of the founders of deep learning. == Early life and education == Aleksey was born in Kobelyaky, Poltava Governorate in a family of teachers. In 1932 he graduated from Electrotechnical college in Kyiv and worked for two years as an engineer on the construction of large power plant in Berezniki. Then in 1938, after graduation from the Leningrad Electrotechnical Institute, Ivakhnenko worked in the All-Union Electrotechnical Institute in Moscow during wartime. There he investigated the problems of automatic control in the laboratory, led by Sergey Lebedev. He continued research in other institutions in Ukraine after return to Kyiv in 1944. In that year he received the Ph.D. degree and later, in 1954 had received D.Sc. degree. In 1964, he was appointed as a Head of the Department of Combined Control Systems at the Institute of Cybernetics. Simultaneously working at first as a Lecturer, and from 1961, as a Professor of Automatic Control and Technical Cybernetics at the Kyiv Polytechnic Institute. == Research == Ivakhnenko is known to be the founder of Inductive modelling, a scientific approach used for pattern recognition and complex systems forecasting. He had used this approach during development of the group method of data handling (GMDH). In 1968 the journal "Avtomatika" had published his article "Group Method of Data Handling – a rival of the method of stochastic approximation", marking the beginning of a new stage in his scientific work. He led the development of this approach, with a professional team of mathematicians and engineers at the Institute of Cybernetics. === Group method of data handling === The GMDH method presents a unique approach to solving problems in artificial intelligence and even a new philosophy to scientific research, which became possible using modern computers. A researcher may not adhere precisely to traditional deductive way of building models "from general theory – to a particular model": monitoring an object, studying its structure, understanding the principles of its operation, developing theory and testing the model of an object. Instead, the new approach is proposed "from specified data – to a general model": after the input of data, a researcher selects a class of models, the type of models-variants generation and sets the criterion for model selection. As most routine work is transferred to a computer, the impact of human influence on the objective result is minimised. In fact, this approach can be considered as one of the implementations of the artificial intelligence thesis, which states that a computer can act as powerful advisor to humans. The development of GMDH consists of a synthesis of ideas from different areas of science: the cybernetic concept of "black box" and the principle of successive genetic selection of pairwise features, Godel's incompleteness theorems and the Gabor's principle of "freedom of decisions choice", the Adhémar's incorrectness and the Beer's principle of external additions. GMDH is the original method for solving problems for structural-parametric identification of models for experimental data under uncertainty. Such a problem occurs in the construction of a mathematical model that approximates the unknown pattern of investigated object or process. It uses information about it that is implicitly contained in data. GMDH differs from other methods of modelling by the active application of the following principles: automatic models generation, inconclusive decisions, and consistent selection by external criteria for finding models of optimal complexity. It had an original multilayered procedure for automatic models structure generation, which imitates the evolutionary process of biological selection with consideration of pairwise successive features. Such procedure is currently used in deep learning networks. To compare and choose optimal models, two or more subsets of a data sample are used. This makes it possible to avoid preliminary assumptions, because sample division implicitly acknowledges different types of uncertainty during the automatic construction of the optimal model. In the early 1980s Ivakhnenko had established an organic analogy between the problem of constructing models for noisy data and signal passing through the channel with noise. This made possible to lay the foundations of the theory of noise-immune modelling. The main result of this theory is that the complexity of optimal predictive model depends on the level of uncertainty in the data: the higher this level (e.g. due to noise) - the simpler must be the optimal model (with less estimated parameters). This initiated the development of the GMDH theory as an inductive method of automatic adaptation of optimal model complexity to the level of information in fuzzy data. Therefore, GMDH is often considered to be the original information technology for knowledge extraction from experimental data. === Results === Alongside to GMDH, Ivakhnenko had developed the following set of results: Principle of construction of self-organizing deep learning networks. Theory of models self-organization according to experimental data. Method of control with forecast optimization. New principles of automatic control of speed for AC and asynchronous electric motors. Theory of invariant systems for adaptive control with compensation of measured disturbances. He had developed the principle of indirect measurement of disturbances, called as "differential fork" that was used later in practice. Principle of combined control (with negative feedback for the controlled variables and positive feedback for the controlled disturbances). A number of such systems, for the speed control of electric motors had been implemented in practice. That proved the practical feasibility of invariant conditions in a combined control systems that unite the advantages of closed systems for control by deviation (high precision) and open systems (performance). The non-searching extreme regulators on the basis of situations recognition. Principle of self-learning pattern recognition. It was demonstrated at first in the cognitive system "Alpha", created under his leadership. Basis for the construction of cybernetic prediction devices. Noise-immune principles of robust modelling for data with noises. Design of multilayered neural networks with active neurons, where each neuron is an algorithm. Ivakhnenko is well known for his achievements in the theory of invariance and theory of combined automatic control systems, that operates on the principle of measured disturbances compensation. He had developed devices and methods for the adaptive control of systems with magnetic amplifiers and motors. He is the author of the first Soviet monograph on engineering cybernetics, which was published worldwide in seven languages. In his study, a further development of the principles of combined control was connected with the implementation of methods of evolutionary self-organisation, pattern recognition and forecasting in control systems. In recent years, his main innovation - the GMDH method - was developed as a method of inductive modelling, complex processes, and systems forecasting. His ideas are utilised now in deep learning networks. The effectiveness of the method was confirmed repeatedly during the solution of real complex problems in ecology, meteorology, economics and technology, which aided increase its popularity among the international scientific community. In parallel, there were conducted developments of evolutionary self-organising algorithms in a related field - clustering problems of pattern recognition. Advances in the modelling of environmental processes reflected in the monographs, economic processes - in the books. The results of exploration of recurrent multilayered GMDH algorithms are described in the books. == Scientific school == From 1963 to 1989 Ivakhnenko was the editor of the specialized scientific journal "Avtomatika" (later "Problems of management and computer science"), that played a crucial role in the formation and development of the Ukrainian school of Inductive modelling. Throughout these years the magazine was translated and reprinted in the United States as "Soviet Automatic Control" (later "Journal of Automation and Information Sciences"). Alongside constant innovation in his field since 1945, Ivakhnenko maintained an active teaching career, at first as the Assistant Professor at the Department of Theoretical Mechanics, and then at the Control Systems faculty. Since 1960 as Professor of the Department of Technical Cybernetics in Kyiv Polytechnic Institute, he contributed lectures to the University and student body, as well as oversaw the work of many graduate students. In 1958-1964 he was an organiser of the All-Union Conferences of Invariance in Kiev, where the development of the invariant control systems theory was restored after prohibition. His inexhaustible enthusiasm helped more than 220 young scientists to prepare and successfully defend their Ph.D. dissertations under his leadership in the KPI and the Institute of Cybernetics and nearly 30 of his students defended their post-doctoral dissertations. Scientific school of Ivakhnenko was and is a real cradle of highly qualified scientific professionals. Furthermore, his students V.M.Kuntsevych, V.I.Kostyuk, V.I.Ivanenko, V.I.Vasiliev, A.A.Pavlov and others had created their own respected scientific schools. Ivakhnenko was a shining example of a scientist, with a keen sense of new and remarkable scientific intuition. Until his last days, he continued to work actively and generously generated original scientific ideas and results. == Awards and honours == Ivakhnenko is the Honorary Scientist of the USSR (1972), two-time winner of the State Prize (1991, 1997) for his works on the theory of invariant automatic systems and set of publications on Information technology in the field of Artificial intelligence. Author of 40 books and over 500 scientific articles. Honorary Doctor of National Technical University "KPI" (2003) and Lviv Polytechnic (2005). He was the Corresponding Member of Academy of Sciences USSR (1961) and Academician of NAS of Ukraine (2003). == Selected works == Ivakhnenko A.G. Heuristic Self-Organization in Problems of Engineering Cybernetics, Automatica, vol.6, 1970 — p. 207-219. Ivakhnenko A.G. Polynomial Theory of Complex Systems, IEEE Transactions on Systems Man and Cybernetics, 4, 1971 — p. 364-378. Ivakhnenko, A.G.; Ivakhnenko, G.A. (1995). "The Review of Problems Solvable by Algorithms of the Group Method of Data Handling (GMDH)" (PDF). Pattern Recognition and Image Analysis. 5 (4): 527–535. CiteSeerX 10.1.1.19.2971. Ivakhnenko, A.G.; Müller, J.-A. (1997). "Recent Developments of Self-Organising Modeling in Prediction And Analysis of Stock Market" (PDF). == References == == External links == Group Method of Data Handling Library of GMDH books and articles GMDH:About the author — Biography and presentations about Aleksey Ivakhnenko.
Wikipedia:Alexey Izmailov#0	Alexey Feridovich Izmailov (Russian: Алексе́й Фери́дович Измаи́лов) (born 1967) is a Russian mathematician who is a professor at the Faculty of Computer Science at the Moscow State University. He defended the thesis Stable methods of finding special solutions of non-linear problems for the degree of Doctor of Physical and Mathematical Sciences in 1998. He has authored 12 books and more than 150 scientific articles. == References == == Bibliography == Grigoriev, Evgeny (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 343–344. ISBN 978-5-211-05838-5. == External links == Home page Scientific works of Alexey Izmailov(in English) Alexey Izmailov publications indexed by Google Scholar
Wikipedia:Alexey Stakhov#0	Alexey Petrovich Stakhov (Russian: Алексей Петрович Стахов Ukrainian: Олексій Петрович Стахов; May 7, 1939 – January 25, 2021) is a Ukrainian mathematician, inventor and engineer, who has made contributions to the theory of Fibonacci numbers and the "Golden Section" and their applications in computer science and measurement theory and technology. Doctor of computer science (1972), professor (1974). Author of over 500 publications, 14 books and 65 international patents. == Biography == Born May 7, 1939, in Partizany, Kherson region, Ukraine, USSR. In 1956 graduated with honours from Rivne village high school. Same year became a student of the Mining Faculty of the Kyiv Polytechnic Institute (now the National Technical University of Ukraine “Kyiv Polytechnic Institute”). In 1959, transferred to the Radio Engineering Faculty of Kharkiv Aviation Institute (now the National Aerospace University of Ukraine). After graduation, worked for two years as an engineer in the Kharkiv Electrical Instrument Design Bureau (now the space technology company “Khartron”). "Khartron" was one of the top secret space companies of the Soviet Union. It was engaged in the research, development and manufacture of automatic control systems for missiles and space craft on board systems. Through working there Stakhov obtained thorough practical engineering experience and published his first scientific papers. Later he worked at the universities of Russia and Ukraine (Kharkiv Institute of Radio Electronics, Taganrog Radio Engineering Institute, Vinnytsia Technical University, Vinnytsia Agricultural University, Vinnytsia Pedagogic University). He was a visiting professor at many universities abroad (Austria, Germany, Libya, Mozambique). Since 2004, he lived and worked in Canada. == Teaching, research, and work == Dean of the Faculty of Computer Engineering of the Kharkiv Institute of Radio Electronics (now, Kharkiv National University of Radio Electronics), 1968–1970 Head of the Department of Informational and Measuring Engineering, Taganrog Radio Engineering Institute, 1971–1977 Head of the Department of Computer Engineering, Vinnytsia Polytechnic Institute (now – Vinnytsia National Technical University), 1977–1988 Director of the Special Design and Technological Bureau "Module" of the Vinnytsia Polytechnic Institute, 1986–1989 Head of the Department of Applied Mathematics and Computing Systems, Vinnytsia Technical University, 1989–1996 Head of the Department of Computer Science, Vinnytsia State Agricultural University (now Vinnytsia National Agrarian University), 1997–2004 Member of the Shevchenko Scientific Society in Canada (2005) Professor Emeritus of Taganrog University of Radio Engineering Since 2003, president of the International Club of the Golden Section, and from 2005, Director of the Golden Section Institute, Academy of Trinitarizm (Russia). Initiated the creation of the so-called "Slavic Golden Group" (Kyiv, 1992) and was Scientific Director of the International Congress on the Mathematics of Harmony (Odesa, 2010). In 2009 he was awarded the Medal "Knight of Arts and Sciences" (Russian Academy of Natural Sciences) == Brief description of research and scientific achievements == Created a new direction in the measurement theory, the algorithmic measurement theory described in his book – A.P. Stakhov. "Introduction into Algorithmic Measurement Theory" (Moscow: Soviet Radio, 1977). Created the theory of the redundant number systems based on the generalised Fibonacci numbers (Fibonacci p-codes) and the generalised golden proportions (codes of the golden p-proportion) The principals of the Fibonacci p-codes and the new computer arithmetic based on them are described in his book – A.P. Stakhov. "Introduction into Algorithmic Measurement Theory" (Moscow: Soviet Radio, 1977). The fundamentals of the theory of the codes of the golden p-proportion are stated in the book – AP Stakhov. The Codes of the Golden Proportion (Moscow, Radio and Communication, 1984). Put forward the concept of "Fibonacci computers". The international priority of this invention was protected by the Soviet Union in an unprecedented patenting operation by Soviet standards, resulting in 65 international patents, covering the territories of the United States, Japan, United Kingdom, France, Germany, Canada and other countries. Elaborated and evolved the concept "Mathematics of Harmony", which goes back in its foundations to Euclid's “Elements” (III c. BC), "Divina Proportione" (Divine proportion) by Luca Pacioli (1509), Harmonice Mundi (Harmony of the World) by Johannes Kepler (1619). For the first time the concept of "Mathematics of Harmony" in relation to the achievements of modern science was presented by Prof. Stakhov in a speech "The Golden Section and Modern Harmony Mathematics," made at the seventh International Conference "Fibonacci Numbers and Their Applications" (Austria, Graz, 1996). The book "The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science" – 748 pages (World Scientific, 2009) is Stakhov's main scientific achievement in this field. Published over 500 scientific works, among them 14 books, 65 international patents, 130 USSR invention certificates. During his work in Canada (2004–2012) published over 30 articles in international journals (Chaos, Solitons & Fractals, Applied Mathematics, Arc Combinatoria, Visual Mathematics, etc.). Alexey Stakhov prepared 30 PhD students, five of Stakhov's PhD students, became Doctors of Sciences. == Work in international universities == The work as "visiting professor": Vienna Technical University (Austria, 1976) University of Jena (Germany, 1986) Dresden Technical University (Germany, 1988) Al Fateh University (Tripoli, Libya, 1995–1997) University of Eduardo Mondlane (Maputo, Mozambique, 1998–2000) == Prizes and awards == Awarded for the best scientific publication by Ministry of Education and Science of Ukraine (1980) Barkhausen's Commemorative Medal issued by the Dresden Technical University as "visiting professor" of Heinrich Barkhausen Department (1988) Professor Emeritus of Taganrog University of Radio Engineering (2004) The honorary title of "Knight of Arts and Sciences" (Russian Academy of Natural Sciences, 2009) The honorary title "Doctor of the Sacred Geometry in Mathematics" (American Society of the Golden Section, 2010) Prof. Alexey Stakhov has been included in the Encyclopedia “Famous Scientists of Russia” == Select scientific publications == Stakhov AP. Introduction into algorithmic measurement theory. Moscow, "Soviet Radio", 1977, 288 p., Circulation 8500 copies.(Russian) Stakhov AP Algorithmic measurement theory. (New in Life, Science and Technology. Series "Mathematics, Cybernetics"). Moscow: "Knowledge", 1979, 64 p. Circulation 38,470 copies. (Ukraininan Ministry of Higher Education awarded the prize for the best scientific publication in 1980) (Russian) Stakhov AP. Codes of the Golden Proportion. Moscow: Radio and Communications, 1984, 152 p., Circulation 10 000 copies. (Russian) Stakhov AP. Algorithmic measurement theory and foundations of computer arithmetic. Journal "Measurement. Control. Automation ", 1981 (Russian) Stakhov AP. The Golden Section in the Measurement Theory. Computers & Mathematics with Applications, 1989, Vol. 17, No 4–6, 613–638. Stakhov A.P. The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers. Kluwer Academic Publishers, 1998 Stakhov A.P. Brousentsov's ternary principle, Bergman's number system and ternary mirror-symmetrical arithmetic. “The Computer Journal” (British Computer Society), 2002. Series of articles (30 articles) on various aspects of the "Mathematics of Harmony." Published at the international journals "Chaos, Solitons & Fractals," «Congressus Numerantium," "Visual Mathematics," "Design & Mathematics," "Applied Mathematics" during 2004–2012. Stakhov AP The Generalized Golden Sections and a New Approach to the Geometric Definition of a Number. Ukrainian Mathematical Journal, 2004 (Russian). Stakhov, AP, Sluchenkova, AA, Shcherbakov, IG. Da Vinci Code and the Fibonacci series. St. Petersburg: Piter, 2006, 320 pp., circulation 8000 copies. Stakhov. AP. The Golden Ratio, Sacred Geometry and the Mathematics of Harmony. The book "Metaphysics. Century XXI ». Moscow, BINOM, 2006, p. 174–215. Stakhov – AP Three "Key" Problems of Mathematics at the Stage of Its Origin and the Mathematics of Harmony as an Alternative Direction in the Development of Mathematics. The book “Totallogy-XXI,” National Academy of Sciences of Ukraine, 2007, p. 274–323. Stakhov A.P. "The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science". International Publisher «World Scientific» (New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai), 2009, p. 748 A. Stakhov, S. Aranson. Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar's Geometry, and Hilbert's Fourth Problem.. Applied Mathematics, 2011, 1 (January), 2 (February), 3 (March) Stakhov AP. The Mathematics of Harmony: from Euclid to Contemporary Mathematics and Computer Science. Electronic Journal "NAUKOVEDENIE ", Vol 4, 2012, p. 105 (Russian) == External links ==
Wikipedia:Alexis Vasseur#0	Alexis Vasseur is a French-American mathematician, specializing in fluid mechanics. He is currently the John T. Stuart III Professor at University of Texas at Austin. In 2015 he was named as a Fellow of the American Mathematical Society. Vasseur earned bachelor's and master's degrees from the Ecole Normale Supérieure in 1993 and 1995, and completed his doctorate from Pierre and Marie Curie University in 1999 under the supervision of Yann Brenier. Before moving to Austin in 2003, he held a permanent research position at CNRS. == References == == External links == Home page
Wikipedia:Alfio Quarteroni#0	Alfio Quarteroni (born 30 May 1952) is an Italian mathematician. He is Professor Emeritus at the Politecnico of Milan (Italy), and Professor Emeritus at the EPFL (Swiss Federal Institute of Technology). He has been the director of the Chair of Modelling and Scientific Computing at the EPFL (Swiss Federal Institute of Technology), Lausanne (Switzerland), from 1998 until 2017. He is the founder (and first director) of MOX at Politecnico of Milan (2002) and MATHICSE at EPFL, Lausanne (2010). He is co-founder (and President) of MOXOFF, a spin-off company at Politecnico of Milan (2010). He is member of the Italian Academy of Science (Accademia Nazionale dei Lincei), the European Academy of Science, the Academia Europaea(Academy of Europe), the Lisbon Academy of Sciences, the Istituto Lombardo Accademia di Scienze e Lettere and the Italian Academy of Engineering and Technology. He is author of 26 books (some of them translated into up to 7 languages), editor of 8 books, author of about 400 papers published in international Scientific Journals and Conference Proceedings, member of the editorial board of 25 International Journals, and Editor in Chief of two-book series published by Springer. He has been an invited or plenary speaker in more than 300 International Conferences and Academic Departments, in particular he has been plenary speaker at ICM 2006 in Madrid. Among his awards and honors are: the NASA Group Achievement Award for the pioneering work in Computational Fluid Dynamics in 1992, the Fanfullino della Riconoscenza 2006, Città di Lodi, the Premio Capo D'Orlando 2006, the Ghislieri prize, 2013, the International Galileo Galilei prize for Sciences 2015. He is recipient of the Galileian Chair from the Scuola Normale Superiore, Pisa, Italy, 2001, doctor Honoris Causa in Naval Engineering from University of Trieste, Italy, 2003, SIAM Fellow (first row) since 2009, IACM (International Association of Computational Mechanics) Fellow since 2004, honorary fellow of ECCOMAS since 2015, the Courant Lectures 2010, the Euler Lecture 2017, the Russell Marker Lectures 2018, the Pedro Nunes Lectures 2018, the Euler Medal from Eccomas (2021–22), the Lagrange Prize from ICIAM (2020–23), the Blaise Pascal Medal in Mathematics in 2024 from the European Academy of Sciences, the Ritz-Galerkin medal from ECCOMAS (2021-24). He has been a member of the IMU Fields Medal Committee for ICM (the International Congress of Mathematicians) 2022. He is the recipient of the ERC Advanced Grant for the project "MATHCARD" in 2008, of two ERC PoC (Proof of Concept) grants: "Math2Ward" in 2012 and "Math4AAARisk" in 2015, and of another ERC Advanced Grant for the project "iHEART" in 2017." His research interests concern Mathematical Modelling, Numerical Analysis, Scientific Computing, Scientific Machine Learning, and application to fluid mechanics, environment, geophysics, medicine, and the improvement of sports performance. He has been the director of 60 Phd students. His research Group at EPFL has contributed to the preliminary design of Solar Impulse, the Swiss long-range experimental solar-powered aircraft project, and has carried out the mathematical simulation for the optimization of performances of the Alinghi yacht, the winner of two editions (2003 and 2007) of the America's Cup. In 2022 he was ranked n.48 in the Top Scientist-Mathematics from research.com, n.1 in Italy. == References ==
Wikipedia:Alfonso Sorrentino (mathematician)#0	Alfonso Sorrentino (Rome, 27 November 1979) is an Italian mathematician, currently full professor of Mathematical analysis at University of Rome Tor Vergata (Italy). His main scientific interests are in the field of dynamical systems, specifically, in the study of Hamiltonian dynamical systems by means of variational methods (Aubry-Mather theory), partial differential equations techniques (weak KAM theory and Hamilton-Jacobi equation) and geometric approaches (symplectic geometry and topology). == Biography and career == Sorrentino was a student of John N. Mather at Princeton University, obtaining his Ph.D. in Mathematics in 2008. After his graduation, he held the roles of junior research fellow at Fondation Sciences Mathématiques de Paris (2008–2009), Herchel-Smith Research Fellow at University of Cambridge (2009–2012), Newton Trust Fellow of Pembroke College, Cambridge (2009–2012), and Researcher at Roma Tre University (2012–2014) == Honors and awards == == Selected writings == A. Sorrentino, Action-minimizing methods in Hamiltonian dynamics. Monograph in the series Mathematical Notes Vol. 50, Princeton University Press, pp. 128, 2015. https://books.google.com/books?id=4PNnBgAAQBAJ V. Kaloshin and A. Sorrentino, On the local Birkhoff conjecture for convex billiards, Annals of Math.(2), 188 (1): 315–380, 2018 https://doi.org/10.4007/annals.2018.188.1.6 G. Huang, V. Kaloshin and A. Sorrentino, On Marked Length Spetrum of Generic Strictly Convex Billiard Tables, Duke Math. Journal, 167 (1): 175 – 209, 2018 https://doi.org/10.1215/00127094-2017-0038 G. Huang, V. Kaloshin and A. Sorrentino, Nearly circular domains which are integrable close to the boundary are ellipses. Geom. and Funct. Analysis, 28 (2): 334–392, 2018 https://doi.org/10.1007/s00039-018-0440-4 A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms. Geom. & Topol., 14 (4): 2383 - 2403, 2010 https://msp.org/gt/2010/14-4/p15.xhtml A. Sorrentino, On the integrability of Tonelli Hamiltonians. Trans. Amer. Math. Soc., 363 (10): 5071 - 5089, 2011 https://www.jstor.org/stable/41307429 L. Butler and A. Sorrentino, Weak Liouville-Arnol'd theorems and their implications. Comm. Math. Phys., 315 (1): 109 – 133, 2012 https://doi.org/10.1007/s00220-012-1536-6 S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Comm. Math. Phys., 354 (2): 775–808, 2017 https://doi.org/10.1007/s00220-017-2900-3 A. Siconolfi and A. Sorrentino, Global results for Eikonal Hamilton-Jacobi equations on networks, Analysis & PDE 11 (1): 171–211, 2018 https://doi.org/10.2140/apde.2018.11.171 == References == == External links == Alfonso Sorrentino publications indexed by Google Scholar University of Rome Tor Vergata, Official website, Alfonso Sorrentino International Mathematics Masters, Alfonso Sorrentino Professor of Mathematical Analysis - University of Rome Tor Vergata, https://www.internationalmathematicsmaster.org/archive/alfonso-sorrentino ANVUR, Alfonso Sorrentino https://www.anvur.it/wp-content/uploads/2022/01/SORRENTINO_Alfonso_CURRICULUM.pdf
Wikipedia:Alfred Errera#0	Alfred Errera (1886 – 1960) was a Belgian mathematician. Errera studied at the Université libre de Bruxelles, where he received his Ph.D. in 1921 with dissertation Du coloriage des cartes et de quelques questions d'analysis situs. In his dissertation he introduced what is now called the Errera graph, which is a counterexample to the validity of the alleged proof of the four color theorem by Alfred Kempe. From 1928 to 1956 he was a professor at the Université libre de Bruxelles. He did research on topology, especially the theory of polyhedra and the Jordan curve theorem. With Théophile Lepage he conducted a seminar on analysis at the University of Brussels. During World War I, he worked on the acoustic source localization of artillery fire and from 1921 to 1938 taught courses on this subject at the Royal Military Academy. Errera was an Invited Speaker at the International Congress of Mathematicians in 1924 in Toronto, in 1932 in Zürich, and in 1936 in Oslo. In 1932 he participated in the Congresul Matematicienilor Români in Drobeta-Turnu Severin. In 1935 he gave a talk at the inaugural Congrés International des Récréations Mathématiques in Brussels. His doctoral students include Guy Hirsch. == Selected publications == Un théorème sur les liaisons, 1923 Periodico matematico, 1927 Analysis situs: un problème d’énumération, Brüssel, M. Lamertin 1931 Sur un problème de géométrie infinitésimale, Brüssel, M. Lamertin 1932 == References == == External links == Despy-Meyer, A.; Halleux, R.; Vandersmissen, J.; Vanpaemel, G. (2001). "Geschiedenis van de wetenschappen in België. 1815–2000 • dbnl". Flemish
Wikipedia:Alfred Frölicher#0	Alfred Frölicher (often misspelled Fröhlicher) was a Swiss mathematician (8 October 8 1927 – 1 July 2010). He was a full professor at the Université de Fribourg (1962–1965), and then at the Université de Genève (1966–1993). He introduced the Frölicher spectral sequence and the Frölicher–Nijenhuis bracket and Frölicher spaces and Frölicher groups. He received his Ph.D. from ETH Zurich in 1954, with thesis Zur Differentialgeometrie der komplexe Strukturen written under the direction of Beno Eckmann and Heinz Hopf. == Publications == Bucher, W.; Frölicher, Alfred (1966), Calculus in vector spaces without norm, Lecture Notes in Mathematics, vol. 30, Berlin, New York: Springer-Verlag, ISBN 978-1-399-86684-2, MR 0213869 Faure, Claude-Alain; Frölicher, Alfred (2000), Modern projective geometry, Mathematics and its Applications, vol. 521, Boston: Kluwer Academic Publishers, ISBN 978-0-7923-6525-9, MR 1783451 Frölicher, Alfred; Kriegl, Andreas (1988), Linear spaces and differentiation theory, Pure and Applied Mathematics (New York), John Wiley & Sons, ISBN 978-0-471-91786-1, MR 0961256 == References ==
Wikipedia:Alfred Klose#0	Wilhelm Rudolf Alfred Klose (September 19, 1895 in Görlitz – February 21, 1953 in Potsdam) was a German applied mathematician and astronomer. == Education and career == Klose studied at University of Breslau and University of Göttingen from 1916 and was an assistant at the observatory in Breslau from 1917. He received his doctorate in astronomy in 1921 in Breslau under Alexander Wilkens (research on the movement of the planet 189 Phthia). After that he was at the University of Greifswald and its observatory, where he habilitated in 1922 and became a privatdozent. In 1923 he habilitated at the Humboldt University of Berlin. In 1924 he became an associate professor of mechanics and theoretical astronomy at University of Latvia in Riga. In 1929 he became an associate professor of astronomy at Humboldt University of Berlin and, after Hilda Geiringer left in 1933, succeeded her as assistant to Theodor Vahlen at the Institute for Applied Mathematics. As an assistant he deputized for Vahlen, who took up a position at the Ministry of Education from 1933 and concentrated on his work there, but had succeeded the expelled Richard von Mises in 1934. In 1937 he became a full professor of applied mathematics and director of the institute. Actually, he had been classified as unsuitable because of his specialization in astronomy. The positive classification by Ludwig Bieberbach and Theodor Vahlen played a role in this – Klose was an active National Socialist. The institute went into decline and even in the Second World War no war-related work was done there - instead, Klose and two assistants were drafted. He was the laboratory manager of a test center of the Army Weapons Office for rocket tests in Gottow and at the Kummersdorf firing range near Luckenwalde. On the other hand, one of his assistants, Karl-Heinz Boseck (exempt from military service due to illness), had a great influence on the mathematics faculty as a former student leader, member of the SS (he set up a department for numerical computing in the Sachsenhausen concentration camp) and fanatical National Socialist of the Second World War. In 1939, Klose was also in talks about the successor to Constantin Caratheodory in Munich (like Karl Strubecker, among others), which, despite positive reports from Vahlen and Bieberbach (who spoke out in favor of staying in Berlin), resulted in negative assessments by Oskar Peron failed. He obtained expert opinions from Ludwig Prandtl and Friedrich Pfeiffer, who were of the opinion that Klose was a particularly unsuspecting ignoramus in the field of applied mathematics. In 1945/46 he was at the successor company of GEMA in Köpenick and from 1946 to 1952 (Operation Ossawakim) as one of the German scientists conscripted into service in the Soviet Union. He was sent to Gorodomlija near Ostashkov along with 169 German rocket scientists and engineers, including Helmut Gröttrup and Fritz Karl Preikschat. In 1952/53 he was a full professor of applied mathematics at the University of Rostock and was commissioned to set up an aeronautical engineering faculty, which only existed there for a short time. Lothar Collatz was one of his doctoral students, but he was the last student of Richard von Mises to complete his work under him and in 1933 he passed the state examination privately with him. In 1936, Klose was one of the editors of the journal German Mathematics. == References == == External links == Sanford L. Segal: Mathematicians under the Nazis, Princeton University Press 2003
Wikipedia:Alfred Korzybski#0	Alfred Habdank Skarbek Korzybski ( ; Polish: [ˈalfrɛt kɔˈʐɨpskʲi]; July 3, 1879 – March 1, 1950) was a Polish-American independent scholar who developed a field called general semantics, which he viewed as both distinct from, and more encompassing than, the field of semantics. He argued that human knowledge of the world is limited both by the human nervous system and the languages humans have developed, and thus no one can have direct access to reality, given that the most we can know is that which is filtered through the brain's responses to reality. His best known dictum is "The map is not the territory". == Early life and career == Born in Warsaw, Vistula Country, which was then part of the Russian Empire, Korzybski belonged to an aristocratic Polish family whose members had worked as mathematicians, scientists, and engineers for generations. He learned the Polish language at home and the Russian language in schools, and having a French and German governess, he became fluent in four languages as a child. Korzybski studied engineering at the Warsaw University of Technology. During the First World War (1914–1918) Korzybski served as an intelligence officer in the Russian Army. After being wounded in a leg and suffering other injuries, he moved to North America in 1916 (first to Canada, then to the United States) to coordinate the shipment of artillery to Russia. He also lectured to Polish-American audiences about the conflict, promoting the sale of war bonds. After the war he decided to remain in the United States, becoming a naturalized citizen in 1940. He met Mira Edgerly, a painter of portraits on ivory, shortly after the 1918 Armistice; They married in January 1919; the marriage lasted until his death. E. P. Dutton published Korzybski's first book, Manhood of Humanity, in 1921. In this work he proposed and explained in detail a new theory of humankind: mankind as a "time-binding" class of life (humans perform time binding by the transmission of knowledge and abstractions through time which become accreted in cultures). In 1925 and 1926, Korzybski observed psychiatric patients at St. Elizabeth's hospital in D.C. under the supervision of William Alanson White. == General semantics == Korzybski's work culminated in the initiation of a discipline that he named general semantics (GS). This should not be confused with semantics. The basic principles of general semantics, which include time-binding, are described in the book Science and Sanity, published in 1933. In 1938, Korzybski founded the Institute of General Semantics in Chicago. The post-World War II housing shortage in Chicago cost him the institute's building lease, so in 1946 he moved the institute to Lakeville, Connecticut, U.S., where he directed it until his death in 1950. Korzybski maintained that humans are limited in what they know by (1) the structure of their nervous systems, and (2) the structure of their languages. Humans cannot experience the world directly, but only through their "abstractions" (nonverbal impressions or "gleanings" derived from the nervous system, and verbal indicators expressed and derived from language). These sometimes mislead us about what is the truth. Our understanding sometimes lacks similarity of structure with what is actually happening. He sought to train our awareness of abstracting, using techniques he had derived from his study of mathematics and science. He called this awareness, this goal of his system, "consciousness of abstracting". His system included the promotion of attitudes such as "I don't know; let's see," in order that we may better discover or reflect on its realities as revealed by modern science. Another technique involved becoming inwardly and outwardly quiet, an experience he termed, "silence on the objective levels". == "To be" == Many devotees and critics of Korzybski reduced his rather complex system to a simple matter of what he said about the verb form "is" of the general verb "to be." His system, however, is based primarily on such terminology as the different "orders of abstraction," and formulations such as "consciousness of abstracting." The contention that Korzybski opposed the use of the verb "to be" would be a profound exaggeration. He thought that certain uses of the verb "to be", called the "is of identity" and the "is of predication", were faulty in structure, e.g., a statement such as, "Elizabeth is a fool" (said of a person named "Elizabeth" who has done something that we regard as foolish). In Korzybski's system, one's assessment of Elizabeth belongs to a higher order of abstraction than Elizabeth herself. Korzybski's remedy was to deny identity; in this example, to be aware continually that "Elizabeth" is not what we call her. We find Elizabeth not in the verbal domain, the world of words, but the nonverbal domain (the two, he said, amount to different orders of abstraction). This was expressed by Korzybski's most famous premise, "the map is not the territory". Note that this premise uses the phrase "is not", a form of "to be"; this and many other examples show that he did not intend to abandon "to be" as such. In fact, he said explicitly that there were no structural problems with the verb "to be" when used as an auxiliary verb or when used to state existence or location. It was even acceptable at times to use the faulty forms of the verb "to be," as long as one was aware of their structural limitations. == Anecdotes == One day, Korzybski was giving a lecture to a group of students, and he interrupted the lesson suddenly in order to retrieve a packet of biscuits, wrapped in white paper, from his briefcase. He muttered that he just had to eat something, and he asked the students on the seats in the front row if they would also like a biscuit. A few students took a biscuit. "Nice biscuit, don't you think," said Korzybski, while he took a second one. The students were chewing vigorously. Then he tore the white paper from the biscuits, in order to reveal the original packaging. On it was a big picture of a dog's head and the words "Dog Cookies." The students looked at the package, and were shocked. Two of them wanted to vomit, put their hands in front of their mouths, and ran out of the lecture hall to the toilet. "You see," Korzybski remarked, "I have just demonstrated that people don't just eat food, but also words, and that the taste of the former is often outdone by the taste of the latter." William Burroughs went to a Korzybski workshop in the autumn of 1939. He was 25 years old, and paid $40. His fellow students—there were 38 in all—included young Samuel I. Hayakawa (later to become a Republican member of the U.S. Senate) and Wendell Johnson (founder of the Monster Study). == Influence == Korzybski's work had a positive reception from a variety of persons in the 1940s and 1950s. Robert A. Heinlein named a character after him in his 1940 short story "Blowups Happen". The science fiction writer A. E. van Vogt based his novel The World of Null-A, published in 1948, on ideas from General Semantics. On March 8, 1949, fellow science-fiction author L. Ron Hubbard wrote to Heinlein referencing Korzybski as an influence on what would become Dianetics: Well, you didn't specify in your book what actual reformation took place in the society to make supermen. Got to thinking about it other day. The system is Excalibur. It makes nul A's. Korzybski's ideas influenced philosopher Alan Watts and physicist Fritjof Capra who used his phrase "the map is not the territory" in lectures and writings "The Tao of Physics", 35th Anniversary Edition. Writer Robert Anton Wilson was also deeply influenced by Korzybski's ideas. The third edition of Science and Sanity states that in World War II the United States Army used Korzybski's system to treat battle fatigue in Europe, under the supervision of Dr. Douglas M. Kelley, who went on to become the psychiatrist in charge of the Nazi war criminals at Nuremberg. Some of the General Semantics tradition was continued by Samuel I. Hayakawa. == Publications == Manhood of Humanity: The Science and Art of Human Engineering. New York: E.P. Dutton. 1921. Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics. Lancaster, Pennsylvania: Science Press Printing Co. 1933. == See also == == References == == Further reading == Kodish, Bruce. 2011. Korzybski: A Biography. Pasadena, CA: Extensional Publishing. ISBN 978-0-9700664-0-4 softcover, 978-09700664-28 hardcover. Kodish, Bruce and Susan Presby Kodish. 2011. Drive Yourself Sane: Using the Uncommon Sense of General Semantics, Third Edition. Pasadena, CA: Extensional Publishing. ISBN 978-0-9700664-1-1 Alfred Korzybski, Manhood of Humanity, foreword by Edward Kasner, notes by M. Kendig, Institute of General Semantics, 1950, hardcover, 2nd edition, 391 pages, ISBN 0-937298-00-X. (Copy of the first edition.) Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics, Alfred Korzybski, preface by Robert P. Pula, Institute of General Semantics, 1994, hardcover, 5th edition, ISBN 0-937298-01-8. (Full text online.) Alfred Korzybski, Collected Writings 1920-1950, Institute of General Semantics, 1990, hardcover, ISBN 0-685-40616-4 Montagu, M. F. A. (1953). Time-binding and the concept of culture. The Scientific Monthly, Vol. 77, No. 3 (Sep., 1953), pp. 148–155. Murray, E. (1950). In memoriam: Alfred H. Korzybski. Sociometry, Vol. 13, No. 1 (Feb., 1950), pp. 76–77. Read, Charlotte (1968). "Alfred Korzybski: His contributions and their historical development". The Polish Review. 13 (2). University of Illinois Press: 5–13. ISSN 0032-2970. JSTOR 25776770. Retrieved January 1, 2025. Rapoport, Anatol (2017). "ALFRED KORZYBSKI JULY 3, 1879 – MARCH 1, 1950 BIOGRAPHICAL SUMMARY". ETC: A Review of General Semantics. 74 (1/2). Institute of General Semantics: 86–92. ISSN 0014-164X. JSTOR 48617411. Retrieved January 9, 2025. Postman, Neil (2003). "Alfred Korzybski". ETC: A Review of General Semantics. 60 (4). Institute of General Semantics: 354–361. ISSN 0014-164X. JSTOR 42578329. Retrieved January 9, 2025. Kodish, Bruce I. (2017). "Young Alfred Korzybski". ETC: A Review of General Semantics. 74 (3/4). Institute of General Semantics: 536–554. ISSN 0014-164X. JSTOR 48617467. Retrieved January 9, 2025. == External links == Works by Alfred Korzybski at Project Gutenberg Works by or about Alfred Korzybski at the Internet Archive Alfred Korzybski and Gestalt Therapy Website Australian General Semantics Society Institute of General Semantics Finding aid to Alfred Korzybski papers at Columbia University. Rare Book & Manuscript Library.
Wikipedia:Alfred Tauber#0	Alfred Tauber (5 November 1866 – 26 July 1942) was a mathematician, known for his contribution to mathematical analysis and to the theory of functions of a complex variable: he is the eponym of an important class of theorems with applications ranging from mathematical and harmonic analysis to number theory. He was born in Austria-Hungary, lived in Vienna, Austria after the dissolution of the empire, and was deported and murdered for being Jewish when the Theresienstadt concentration camp was emptied of Jews in 1942. == Life and academic career == Born in Pressburg, Kingdom of Hungary, Austrian Empire (now Bratislava, Slovakia), he began studying mathematics at Vienna University in 1884, obtained his Ph.D. in 1889, and his habilitation in 1891. Starting from 1892, he worked as chief mathematician at the Phönix insurance company until 1908, when he became an a.o. professor at the University of Vienna, though, already from 1901, he had been honorary professor at TU Vienna and director of its insurance mathematics chair. In 1933, he was awarded the Grand Decoration of Honour in Silver for Services to the Republic of Austria, and retired as emeritus extraordinary professor. However, he continued lecturing as a privatdozent until 1938, when he was forced to resign as a consequence of the "Anschluss". On 28–29 June 1942, he was deported with transport IV/2, č. 621 to Theresienstadt, where he was murdered on 26 July 1942. == Work == Pinl & Dick (1974, p. 202) list 35 publications in the bibliography appended to his obituary, and also a search performed on the "Jahrbuch über die Fortschritte der Mathematik" database results in a list 35 mathematical works authored by him, spanning a period of time from 1891 to 1940. However, Hlawka (2007) cites two papers on actuarial mathematics which do not appear in these two bibliographical lists and Binder's bibliography of Tauber's works (1984, pp. 163–166), while listing 71 entries including the ones in the bibliography of Pinl & Dick (1974, p. 202) and the two cited by Hlawka, does not includes the short note (Tauber 1895) so the exact number of his works is not known. According to Hlawka (2007), his scientific research can be divided into three areas: the first one comprises his work on the theory of functions of a complex variable and on potential theory, the second one includes works on linear differential equations and on the Gamma function, while the last one includes his contributions to actuarial science. Pinl & Dick (1974, p. 202) give a more detailed list of research topics Tauber worked on, though it is restricted to mathematical analysis and geometric topics: some of them are infinite series, Fourier series, spherical harmonics, the theory of quaternions, analytic and descriptive geometry. Tauber's most important scientific contributions belong to the first of his research areas, even if his work on potential theory has been overshadowed by the one of Aleksandr Lyapunov. === Tauberian theorems === His most important article is (Tauber 1897). In this paper, he succeeded in proving a converse to Abel's theorem for the first time: this result was the starting point of numerous investigations, leading to the proof and to applications of several theorems of such kind for various summability methods. The statement of these theorems has a standard structure: if a series ∑ an is summable according to a given summability method and satisfies an additional condition, called "Tauberian condition", then it is a convergent series. Starting from 1913 onward, G. H. Hardy and J. E. Littlewood used the term Tauberian to identify this class of theorems. Describing with a little more detail Tauber's 1897 work, it can be said that his main achievements are the following two theorems: Tauber's first theorem. If the series ∑ an is Abel summable to sum s, i.e. limx→ 1− ∑+∞n=0 an x n = s, and if an = ο(n−1), then ∑ ak converges to s. This theorem is, according to Korevaar (2004, p. 10), the forerunner of all Tauberian theory: the condition an = ο(n−1) is the first Tauberian condition, which later had many profound generalizations. In the remaining part of his paper, by using the theorem above, Tauber proved the following, more general result: Tauber's second theorem. The series ∑ an converges to sum s if and only if the two following conditions are satisfied: ∑ an is Abel summable and ∑nk=1 k ak = ο(n). This result is not a trivial consequence of Tauber's first theorem. The greater generality of this result with respect to the former one is due to the fact it proves the exact equivalence between ordinary convergence on one side and Abel summability (condition 1) jointly with Tauberian condition (condition 2) on the other. Chatterji (1984, pp. 169–170) claims that this latter result must have appeared to Tauber much more complete and satisfying respect to the former one as it states a necessary and sufficient condition for the convergence of a series while the former one was simply a stepping stone to it: the only reason why Tauber's second theorem is not mentioned very often seems to be that it has no profound generalization as the first one has, though it has its rightful place in all detailed developments of summability of series. === Contributions to the theory of Hilbert transform === Frederick W. King (2009, p. 3) writes that Tauber contributed at an early stage to theory of the now called "Hilbert transform", anticipating with his contribution the works of Hilbert and Hardy in such a way that the transform should perhaps bear their three names. Precisely, Tauber (1891) considers the real part φ and imaginary part ψ of a power series f, f ( z ) = ∑ k = 1 + ∞ c k z k = φ ( θ ) + i ψ ( θ ) {\displaystyle f(z)=\sum _{k=1}^{+\infty }c_{k}z^{k}=\varphi (\theta )+\mathrm {i} \psi (\theta )} where z = re iθ with r = \| z \| being the absolute value of the given complex variable, ck r k = ak + ibk for every natural number k, φ(θ) = ∑+∞k=1 akcos(kθ) − bksin(kθ) and ψ(θ) = ∑+∞k=1 aksin(kθ) + bkcos(kθ) are trigonometric series and therefore periodic functions, expressing the real and imaginary part of the given power series. Under the hypothesis that r is less than the convergence radius Rf of the power series f, Tauber proves that φ and ψ satisfy the two following equations: (1) φ ( θ ) = 1 2 π ∫ 0 π { ψ ( θ + ϕ ) − ψ ( θ − ϕ ) } cot ⁡ ( ϕ 2 ) d ϕ {\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\psi (\theta +\phi )-\psi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\,\mathrm {d} \phi } (2) ψ ( θ ) = − 1 2 π ∫ 0 π { φ ( θ + ϕ ) − φ ( θ − ϕ ) } cot ⁡ ( ϕ 2 ) d ϕ {\displaystyle \psi (\theta )=-{\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\varphi (\theta +\phi )-\varphi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\mathrm {d} \phi } Assuming then r = Rf, he is also able to prove that the above equations still hold if φ and ψ are only absolutely integrable: this result is equivalent to defining the Hilbert transform on the circle since, after some calculations exploiting the periodicity of the functions involved, it can be proved that (1) and (2) are equivalent to the following pair of Hilbert transforms: φ ( θ ) = 1 2 π ∫ − π π ψ ( ϕ ) cot ⁡ ( θ − ϕ 2 ) d ϕ ψ ( θ ) = 1 2 π ∫ − π π φ ( ϕ ) cot ⁡ ( θ − ϕ 2 ) d ϕ {\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\psi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi \qquad \psi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\varphi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi } Finally, it is perhaps worth pointing out an application of the results of (Tauber 1891), given (without proof) by Tauber himself in the short research announce (Tauber 1895): the complex valued continuous function φ(θ) + iψ(θ) defined on a given circle is the boundary value of a holomorphic function defined in its open disk if and only if the two following conditions are satisfied the function [φ(θ − α) − φ(θ + α)]/α is uniformly integrable in every neighborhood of the point α = 0, and the function ψ(θ) satisfies (2). == Selected publications == Tauber, Alfred (1891), "Über den Zusammenhang des reellen und imaginären Theiles einer Potenzreihe" [On the relation between real and imaginary part of a power series], Monatshefte für Mathematik und Physik, II: 79–118, doi:10.1007/bf01691828, JFM 23.0251.01, S2CID 120241651. Tauber, Alfred (1895), "Ueber die Werte einer analytischen Function längs einer Kreislinie" [On the values of an analytic function along a circular perimeter], Jahresbericht der Deutschen Mathematiker-Vereinigung, 4: 115, archived from the original on 2015-07-01, retrieved 2014-07-16. Tauber, Alfred (1897), "Ein Satz aus der Theorie der unendlichen Reihen" [A theorem about infinite series], Monatshefte für Mathematik und Physik, VIII: 273–277, doi:10.1007/BF01696278, JFM 28.0221.02, S2CID 120692627. Tauber, Alfred (1898), "Über einige Sätze der Potentialtheorie" [Some theorems of potential theory], Monatshefte für Mathematik und Physik, IX: 79–118, doi:10.1007/BF01707858, JFM 29.0654.02, S2CID 124400762. Tauber, Alfred (1920), "Über konvergente und asymptotische Darstellung des Integrallogarithmus" [On convergent and asymptotic representation of the logarithmic integral function], Mathematische Zeitschrift, 8 (1–2): 52–62, doi:10.1007/bf01212858, JFM 47.0329.01, S2CID 119967249. Tauber, Alfred (1922), "Über die Umwandlung von Potenzreihen in Kettenbrüche" [On the conversion of power series into continued fractions], Mathematische Zeitschrift, 15: 66–80, doi:10.1007/bf01494383, JFM 48.0236.01, S2CID 122501264. == See also == Actuarial science Hardy–Littlewood tauberian theorem Summability theory == Notes == == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Alfred Tauber", MacTutor History of Mathematics Archive, University of St Andrews Alfred Tauber at encyclopedia.com Alfred Tauber at the Mathematics Genealogy Project
Wikipedia:Alfredo Noel Iusem#0	Alfredo Noel Iusem (born 10 November 1949, in Buenos Aires) is an Argentine-born Brazilian mathematician working on mathematical optimization. He earned his Ph.D. from Stanford University in 1981 under the supervision of George Bernard Dantzig. He is a recipient of Brazil's National Order of Scientific Merit in mathematics. Since 2001, he has been a member of the Brazilian Academy of Sciences. Regina Burachik was a student of his at Instituto Nacional de Matemática Pura e Aplicada. == Selected publications == Burachik, Regina S.; Iusem, Alfredo N. (1998). "A generalized proximal point algorithm for the variational inequality problem in a Hilbert space". SIAM Journal on Optimization. 8 (1): 197–216. doi:10.1137/S1052623495286302. Burachik, Regina S.; Iusem, Alfredo N.; Svaiter, B. F. (1997). "Enlargement of monotone operators with applications to variational inequalities". Set-Valued Analysis and Variational Analysis. 5 (2): 159–180. doi:10.1023/A:1008615624787. Burachik, Regina S.; Svaiter, B. F.; Teboulle, Marc (1994). "Entropy-like proximal methods in convex programming". Mathematics of Operations Research. 19 (4): 790–814. doi:10.1287/moor.19.4.790. == References ==
Wikipedia:Algebra#0	Algeria, officially the People's Democratic Republic of Algeria, is a country in the Maghreb region of North Africa. It is bordered to the northeast by Tunisia; to the east by Libya; to the southeast by Niger; to the southwest by Mali, Mauritania, and Western Sahara; to the west by Morocco; and to the north by the Mediterranean Sea. The capital and largest city is Algiers, located in the far north on the Mediterranean coast. Inhabited since prehistory, Algeria has been at the crossroads of numerous cultures and civilisations, including the Phoenicians, Numidians, Romans, Vandals, and Byzantine Greeks. Its modern identity is rooted in centuries of Arab Muslim migration waves since the seventh century and the subsequent Arabisation of the indigenous populations. Following a succession of Islamic Arab and Berber dynasties between the eighth and 15th centuries, the Regency of Algiers was established in 1516 as a largely independent tributary state of the Ottoman Empire. After nearly three centuries as a major power in the Mediterranean, the country was invaded by France in 1830 and formally annexed in 1848, though it was not fully conquered and pacified until 1903. French rule brought mass European settlement that displaced the local population, which was reduced by up to one-third due to warfare, disease, and starvation. The Sétif and Guelma massacre in 1945 catalysed local resistance that culminated in the outbreak of the Algerian War in 1954. Algeria gained its independence in 1962. The country descended into a bloody civil war from 1992 to 2002. Spanning 2,381,741 square kilometres (919,595 sq mi), Algeria is the world's tenth-largest nation by area, and the largest nation in Africa. It has a semi-arid climate, with the Sahara desert dominating most of the territory except for its fertile and mountainous north, where most of the population is concentrated. With a population of 44 million, Algeria is the tenth-most populous country in Africa, and the 33rd-most populous country in the world. Algeria's official languages are Arabic and Tamazight; French is used in media, education, and certain administrative matters, but it has no official status. The vast majority of the population speak the Algerian dialect of Arabic. Most Algerians are Arabs, with Berbers forming a sizeable minority. Sunni Islam is the official religion and practised by 99 percent of the population. Algeria is a semi-presidential republic composed of 58 provinces (wilayas) and 1,541 communes. It is a regional power in North Africa and a middle power in global affairs. As of 2025, the country has the highest Human Development Index in continental Africa, and one of the largest economies in Africa, due mostly to its large petroleum and natural gas reserves, which are the sixteenth and ninth-largest in the world, respectively. Sonatrach, the national oil company, is the largest company in Africa and a major supplier of natural gas to Europe. The Algerian military is one of the largest in Africa, with the highest defence budget on the continent and among the highest in the world (ranks 22nd globally). Algeria is a member of the African Union, the Arab League, the OIC, OPEC, the United Nations, and the Arab Maghreb Union, of which it is a founding member. == Name == Different forms of the name Algeria include: Arabic: الجزائر, romanized: al-Jazāʾir, Algerian Arabic: دزاير, romanized: dzāyer, French: l'Algérie. The country's full name is officially the People's Democratic Republic of Algeria (Arabic: الجمهورية الجزائرية الديمقراطية الشعبية, romanized: al-Jumhūriyah al-Jazāʾiriyah ad-Dīmuqrāṭiyah ash‑Shaʿbiyah; French: République algérienne démocratique et populaire, abbr. RADP; Berber Tifinagh: ⵜⴰⴳⴷⵓⴷⴰ ⵜⴰⵣⵣⴰⵢⵔⵉⵜ ⵜⴰⵎⴰⴳⴷⴰⵢⵜ ⵜⴰⵖⴻⵔⴼⴰⵏⵜ, Berber Latin alphabet: Tagduda tazzayrit tamagdayt taɣerfant). === Etymology === Algeria's name derives from the city of Algiers, which in turn derives from the Arabic al-Jazāʾir (الجزائر, 'the islands'), referring to four small islands off its coast, a truncated form of the older Jazāʾir Banī Mazghanna (جزائر بني مزغنة, 'islands of Bani Mazghanna'). The name was given by Buluggin ibn Ziri after he established the city on the ruins of the Phoenician city of Icosium in 950. It was employed by medieval geographers such as Muhammad al-Idrisi and Yaqut al-Hamawi. Algeria took its name from the Regency of Algeria or Regency of Algiers, when Ottoman rule was established in the central Maghreb in early 16th century. This period saw the installation of a political and administrative organisation which participated in the establishment of the Watan el djazâïr (وطن الجزائر, 'country of Algiers') and the definition of its borders with its neighboring entities on the east and west. The Ottoman Turks who settled in Algeria referred both to themselves and the peoples as "Algerians". Acting as a central military and political authority in the regency, the Ottoman Turks shaped the modern political identity of Algeria as a state possessing all the attributes of sovereign independence, despite still being nominally subject to the Ottoman sultan. Algerian nationalist, historian and statesman Ahmed Tewfik El Madani regarded the regency as the "first Algerian state" and the "Algerian Ottoman republic". == History == === Prehistory and ancient history === Around ~1.8-million-year-old stone artifacts from Ain Hanech (Algeria) were considered to represent the oldest archaeological materials in North Africa. Stone artifacts and cut-marked bones that were excavated from two nearby deposits at Ain Boucherit are estimated to be ~1.9 million years old, and even older stone artifacts to be as old as ~2.4 million years. Hence, the Ain Boucherit evidence shows that ancestral hominins inhabited the Mediterranean fringe in northern Africa much earlier than previously thought. The evidence strongly argues for early dispersal of stone tool manufacture and use from East Africa, or a possible multiple-origin scenario of stone technology in both East and North Africa. Neanderthal tool makers produced hand axes in the Levalloisian and Mousterian styles (43,000 BC) similar to those in the Levant. Algeria was the site of the highest state of development of Middle Paleolithic Flake tool techniques. Tools of this era, starting about 30,000 BC, are called Aterian (after the archaeological site of Bir el Ater, south of Tebessa). The earliest blade industries in North Africa are called Iberomaurusian (located mainly in the Oran region). This industry appears to have spread throughout the coastal regions of the Maghreb between 15,000 and 10,000 BC. Neolithic civilisation (animal domestication and agriculture) developed in the Saharan and Mediterranean Maghreb perhaps as early as 11,000 BC or as late as between 6000 and 2000 BC. This life, richly depicted in the Tassili n'Ajjer paintings, predominated in Algeria until the classical period. The mixture of peoples of North Africa coalesced eventually into a distinct native population that came to be called Berbers, who are the indigenous peoples of northern Africa. From their principal center of power at Carthage, the Carthaginians expanded and established small settlements along the North African coast; by 600 BC, a Phoenician presence existed at Tipasa, east of Cherchell, Hippo Regius (modern Annaba) and Rusicade (modern Skikda). These settlements served as market towns as well as anchorages. As Carthaginian power grew, its impact on the indigenous population increased dramatically. Berber civilisation was already at a stage in which agriculture, manufacturing, trade, and political organisation supported several states. Trade links between Carthage and the Berbers in the interior grew, but territorial expansion also resulted in the enslavement or military recruitment of some Berbers and in the extraction of tribute from others. By the early 4th century BC, The north is divided into two Masaesyli kingdom in west led by Syphax and Massylii kingdom in east. Berbers formed the single largest element of the Carthaginian army. In the Revolt of the Mercenaries, Berber soldiers rebelled from 241 to 238 BC after being unpaid following the defeat of Carthage in the First Punic War. They succeeded in obtaining control of much of Carthage's North African territory, and they minted coins bearing the name Libyan, used in Greek to describe natives of North Africa. The Carthaginian state declined because of successive defeats by the Romans in the Punic Wars. In 146 BC the city of Carthage was destroyed. As Carthaginian power waned, the influence of Berber leaders in the hinterland grew. By the 2nd century BC, several large but loosely administered Berber kingdoms had emerged. Two of them were established in Numidia, behind the coastal areas controlled by Carthage. West of Numidia lay Mauretania, which extended across the Moulouya River in modern-day Morocco to the Atlantic Ocean. The high point of Berber civilisation, unequalled until the coming of the Almohads and Almoravids more than a millennium later, was reached during the reign of Masinissa in the 2nd century BC. After Masinissa's death in 148 BC, the Berber kingdoms were divided and reunited several times. Masinissa's line survived until 24 AD, when the remaining Berber territory was annexed to the Roman Empire. For several centuries Algeria was ruled by the Romans, who founded many colonies in the region. Algeria is home to the second-largest number of Roman sites and remains after Italy. Rome, after getting rid of its powerful rival Carthage in the year 146 BC, decided a century later to include Numidia to become the new master of North Africa. They built more than 500 cities. Like the rest of North Africa, Algeria was one of the breadbaskets of the empire, exporting cereals and other agricultural products. Saint Augustine was the bishop of Hippo Regius (modern-day Annaba, Algeria), located in the Roman province of Africa. The Germanic Vandals of Geiseric moved into North Africa in 429, and by 435 controlled coastal Numidia. They did not make any significant settlement on the land, as they were harassed by local tribes. In fact, by the time the Byzantines arrived Leptis Magna was abandoned and the Msellata region was occupied by the indigenous Laguatan who had been busy facilitating an Amazigh political, military and cultural revival. Furthermore, during the rule of the Romans, Byzantines, Vandals, Carthaginians, and Ottomans the Berber people were the only or one of the few in North Africa who remained independent. The Berber people were so resistant that even during the Muslim conquest of North Africa they still had control and possession over their mountains. The collapse of the Western Roman Empire led to the establishment of a native Kingdom based in Altava (modern-day Algeria) known as the Mauro-Roman Kingdom. It was succeeded by another Kingdom based in Altava, the Kingdom of Altava. During the reign of Kusaila its territory extended from the region of modern-day Fez in the west to the western Aurès and later Kairaouan and the interior of Ifriqiya in the east. === Middle Ages === After negligible resistance from the locals, Muslim Arabs of the Umayyad Caliphate conquered Algeria in the early 8th century. Large numbers of the indigenous Berber people converted to Islam. Christians, Berber and Latin speakers remained in the great majority in Tunisia until the end of the 9th century and Muslims only became a vast majority some time in the 10th. After the fall of the Umayyad Caliphate, numerous local dynasties emerged, including the Rustamids, Aghlabids, Fatimids, Zirids, Hammadids, Almoravids, Almohads and the Zayyanids. The Christians left in three waves: after the initial conquest, in the 10th century and the 11th. The last were evacuated to Sicily by the Normans and the few remaining died out in the 14th century. During the Middle Ages, North Africa was home to many great scholars, saints and sovereigns including Judah Ibn Quraysh, the first grammarian to mention Semitic and Berber languages, the great Sufi masters Sidi Boumediene (Abu Madyan) and Sidi El Houari, and the Emirs Abd Al Mu'min and Yāghmūrasen. It was during this time that the Fatimids or children of Fatima, daughter of Muhammad, came to the Maghreb. These "Fatimids" went on to found a long lasting dynasty stretching across the Maghreb, Hejaz and the Levant, boasting a secular inner government, as well as a powerful army and navy, made up primarily of Arabs and Levantines extending from Algeria to their capital state of Cairo. The Fatimid caliphate began to collapse when its governors the Zirids seceded. To punish them the Fatimids sent the Arab Banu Hilal and Banu Sulaym against them. The resultant war is recounted in the epic Tāghribāt. In Al-Tāghrībāt the Amazigh Zirid Hero Khālīfā Al-Zānatī asks daily, for duels, to defeat the Hilalan hero Ābu Zayd al-Hilalī and many other Arab knights in a string of victories. The Zirids, however, were ultimately defeated ushering in an adoption of Arab customs and culture. The indigenous Amazigh tribes, however, remained largely independent, and depending on tribe, location and time controlled varying parts of the Maghreb, at times unifying it (as under the Fatimids). The Fatimid Islamic state, also known as Fatimid Caliphate made an Islamic empire that included North Africa, Sicily, Palestine, Jordan, Lebanon, Syria, Egypt, the Red Sea coast of Africa, Tihamah, Hejaz and Yemen. Caliphates from Northern Africa traded with the other empires of their time, as well as forming part of a confederated support and trade network with other Islamic states during the Islamic Era. The Berber people historically consisted of several tribes. The two main branches were the Botr and Barnès tribes, who were divided into tribes, and again into sub-tribes. Each region of the Maghreb contained several tribes (for example, Sanhadja, Houara, Zenata, Masmouda, Kutama, Awarba, and Berghwata). All these tribes made independent territorial decisions. Several Amazigh dynasties emerged during the Middle Ages in the Maghreb and other nearby lands. Ibn Khaldun provides a table summarising the Amazigh dynasties of the Maghreb region, the Zirid, Ifranid, Maghrawa, Almoravid, Hammadid, Almohad, Merinid, Abdalwadid, Wattasid, Meknassa and Hafsid dynasties. Both of the Hammadid and Zirid empires as well as the Fatimids established their rule in all of the Maghreb countries. The Zirids ruled land in what is now Algeria, Tunisia, Morocco, Libya, Spain, Malta and Italy. The Hammadids captured and held important regions such as Ouargla, Constantine, Sfax, Susa, Algiers, Tripoli and Fez establishing their rule in every country in the Maghreb region. The Fatimids which was created and established by the Kutama Berbers conquered all of North Africa as well as Sicily and parts of the Middle East. Following the Berber revolt numerous independent states emerged across the Maghreb. In Algeria the Rustamid Kingdom was established. The Rustamid realm stretched from Tafilalt in Morocco to the Nafusa mountains in Libya including south, central and western Tunisia therefore including territory in all of the modern day Maghreb countries, in the south the Rustamid realm expanded to the modern borders of Mali and included territory in Mauritania. Once extending their control over all of the Maghreb, part of Spain and briefly over Sicily, originating from modern Algeria, the Zirids only controlled modern Ifriqiya by the 11th century. The Zirids recognised nominal suzerainty of the Fatimid caliphs of Cairo. El Mu'izz the Zirid ruler decided to end this recognition and declared his independence. The Zirids also fought against other Zenata Kingdoms, for example the Maghrawa, a Berber dynasty originating from Algeria and which at one point was a dominant power in the Maghreb ruling over much of Morocco and western Algeria including Fez, Sijilmasa, Aghmat, Oujda, most of the Sous and Draa and reaching as far as M'sila and the Zab in Algeria. As the Fatimid state was at the time too weak to attempt a direct invasion, they found another means of revenge. Between the Nile and the Red Sea were living Bedouin nomad tribes expelled from Arabia for their disruption and turbulency. The Banu Hilal and the Banu Sulaym for example, who regularly disrupted farmers in the Nile Valley since the nomads would often loot their farms. The then Fatimid vizier decided to destroy what he could not control, and broke a deal with the chiefs of these Bedouin tribes. The Fatimids even gave them money to leave. Whole tribes set off with women, children, elders, animals and camping equipment. Some stopped on the way, especially in Cyrenaica, where they are still one of the essential elements of the settlement but most arrived in Ifriqiya by the Gabes region, arriving 1051. The Zirid ruler tried to stop this rising tide, but with each encounter, the last under the walls of Kairouan, his troops were defeated and the Arabs remained masters of the battlefield. The Arabs usually did not take control over the cities, instead looting them and destroying them. The invasion kept going, and in 1057 the Arabs spread on the high plains of Constantine where they encircled the Qalaa of Banu Hammad (capital of the Hammadid Emirate), as they had done in Kairouan a few decades ago. From there they gradually gained the upper Algiers and Oran plains. Some of these territories were forcibly taken back by the Almohads in the second half of the 12th century. The influx of Bedouin tribes was a major factor in the linguistic, cultural Arabisation of the Maghreb and in the spread of nomadism in areas where agriculture had previously been dominant. Ibn Khaldun noted that the lands ravaged by the Banu Hilal tribes had become completely arid desert. The Almohads originating from modern day Morocco, although founded by a man originating from modern day Algeria known as Abd al-Mu'min would soon take control over the Maghreb. During the time of the Almohad Dynasty Abd al-Mu'min's tribe, the Koumïa, were the main supporters of the throne and the most important body of the empire. Defeating the weakening Almoravid Empire and taking control over Morocco in 1147, they pushed into Algeria in 1152, taking control over Tlemcen, Oran, and Algiers, wrestling control from the Hilian Arabs, and by the same year they defeated Hammadids who controlled Eastern Algeria. Following their decisive defeat at the Battle of Las Navas de Tolosa in 1212, the Almohads began to collapse. In 1235, the governor of modern-day western Algeria, Yaghmurasen Ibn Zyan, declared independence and established the Kingdom of Tlemcen under the Zayyanid dynasty. After warring with the Almohads for 13 years, the Zayyanids decisively defeated them in 1248 by ambushing and killing the Almohad caliph near Oujda. The Zayyanids retained control over much of Algeria for the next three centuries. While eastern Algeria largely fell under the Hafsid dynasty, the Emirate of Bejaia, which encompassed the Hafsid territories in Algeria, was at times independent from central Tunisian authority. At their peak, the Zayyanids held Morocco as a western vassal expanded eastward as far as Tunis, which was captured during the reign of Abu Tashfin. Following several conflicts with local Barbary pirates sponsored by the Zayyanid sultans, Spain launched a campaign to invade Algeria and defeat the Kingdom of Tlemcen. In 1505, Spanish forces invaded and captured Mers el Kébir, and in 1509, they conquered Oran after a deadly siege. Following their decisive victories over the Algerians in the western-coastal areas of Algeria, the Spanish expanded their campaign across the western Algerian coast. In 1510, they captured Bejaia after a major siege, launched an assault on Algiers, and besieged Tlemcen. In 1511, they seized Cherchell and Jijel, and attacked Mostaganem, which they failed to conquer but succeeded in forcing into tribute. === Early modern era === In 1516, the Turkish privateer brothers Aruj and Hayreddin Barbarossa, who operated successfully under the Hafsids, moved their base of operations to Algiers. They succeeded in conquering Jijel and Algiers from the Spaniards with help from the locals who saw them as liberators from the Christians, but the brothers eventually assassinated the local noble Salim al-Tumi and took control over the city and the surrounding regions. Their state is known as the Regency of Algiers. When Aruj was killed in 1518 during his invasion of Tlemcen, Hayreddin succeeded him as military commander of Algiers. The Ottoman sultan gave him the title of beylerbey and a contingent of some 2,000 janissaries. With the aid of this force and native Algerians, Hayreddin conquered the whole area between Constantine and Oran (although the city of Oran remained in Spanish hands until 1792). The next beylerbey was Hayreddin's son Hasan, who assumed the position in 1544. He was a Kouloughli or of mixed origins, as his mother was an Algerian Mooresse. Until 1587 Beylerbeylik of Algiers was governed by Beylerbeys who served terms with no fixed limits. Subsequently, with the institution of a regular administration, governors with the title of pasha ruled for three-year terms. The pasha was assisted by an autonomous janissary unit, known in Algeria as the Ojaq who were led by an agha. Discontent among the ojaq rose in the mid-1600s because they were not paid regularly, and they repeatedly revolted against the pasha. As a result, the agha charged the pasha with corruption and incompetence and seized power in 1659. Plague had repeatedly struck the cities of North Africa. Algiers lost between 30,000 and 50,000 inhabitants to the plague in 1620–21, and had high fatalities in 1654–57, 1665, 1691 and 1740–42. The Barbary pirates preyed on Christian and other non-Islamic shipping in the western Mediterranean Sea. The pirates often took the passengers and crew on the ships and sold them or used them as slaves. They also did a brisk business in ransoming some of the captives. According to Robert Davis, from the 16th to 19th century, pirates captured 1 million to 1.25 million Europeans as slaves. They often made raids on European coastal towns to capture Christian slaves to sell at slave markets in North Africa and other parts of the Ottoman Empire. In 1544, for example, Hayreddin Barbarossa captured the island of Ischia, taking 4,000 prisoners, and enslaved some 9,000 inhabitants of Lipari, almost the entire population. In 1551, the Ottoman governor of Algiers, Turgut Reis, enslaved the entire population of the Maltese island of Gozo. Barbary pirates often attacked the Balearic Islands. The threat was so severe that residents abandoned the island of Formentera. The introduction of broad-sail ships from the beginning of the 17th century allowed them to branch out into the Atlantic. In July 1627 two pirate ships from Algiers under the command of Dutch pirate Jan Janszoon sailed as far as Iceland, raiding and capturing slaves. Two weeks earlier another pirate ship from Salé in Morocco had also raided in Iceland. Some of the slaves brought to Algiers were later ransomed back to Iceland, but some chose to stay in Algeria. In 1629, pirate ships from Algeria raided the Faroe Islands. In 1659, the Janissaries stationed in Algiers, also known commonly as the Odjak of Algiers; and the Reis or the company of corsair captains rebelled, they removed the Ottoman viceroy from power, and placed one of its own in power. The new leader received the title of "Agha" then "Dey" in 1671, and the right to select passed to the divan, a council of some sixty military senior officers. Thus Algiers became a sovereign military republic. It was at first dominated by the odjak; but by the 18th century, it had become the dey's instrument. Although Algiers remained nominally part of the Ottoman Empire, in reality they acted independently from the rest of the Empire, and often had wars with other Ottoman subjects and territories such as the Beylik of Tunis. The dey was in effect a constitutional autocrat. The dey was elected for a life term, but in the 159 years (1671–1830) that the system was in place, fourteen of the twenty-nine deys were assassinated. Despite usurpation, military coups and occasional mob rule, the day-to-day operation of the Deylikal government was remarkably orderly. Although the regency patronised the tribal chieftains, it never had the unanimous allegiance of the countryside, where heavy taxation frequently provoked unrest. Autonomous tribal states were tolerated, and the regency's authority was seldom applied in the Kabylia, although in 1730 the Regency was able to take control over the Kingdom of Kuku in western Kabylia. Many cities in the northern parts of the Algerian desert paid taxes to Algiers or one of its Beys. Barbary raids in the Mediterranean continued to attack Spanish merchant shipping, and as a result, the Spanish Empire launched an invasion in 1775, then the Spanish Navy bombarded Algiers in 1783 and 1784. For the attack in 1784, the Spanish fleet was to be joined by ships from such traditional enemies of Algiers as Naples, Portugal and the Knights of Malta. Over 20,000 cannonballs were fired, but all these military campaigns were doomed and Spain had to ask for peace in 1786 and paid 1 million pesos to the Dey. In 1792, Algiers took back Oran and Mers el Kébir, the two last Spanish strongholds in Algeria. In the same year, they conquered the Moroccan Rif and Oujda, which they then abandoned in 1795. In the 19th century, Algerian pirates forged affiliations with Caribbean powers, paying a "license tax" in exchange for safe harbor of their vessels. Attacks by Algerian pirates on American merchantmen resulted in the First and Second Barbary Wars, which ended the attacks on U.S. ships in 1815. A year later, a combined Anglo-Dutch fleet, under the command of Lord Exmouth bombarded Algiers to stop similar attacks on European fishermen. These efforts proved successful, although Algerian piracy would continue until the French conquest in 1830. === French colonisation (1830–1962) === Under the pretext of a slight to their consul, the French invaded and captured Algiers in 1830. According to several historians, the methods used by the French to establish control over Algeria reached genocidal proportions. Historian Ben Kiernan wrote on the French conquest of Algeria: "By 1875, the French conquest was complete. The war had killed approximately 825,000 indigenous Algerians since 1830". French losses from 1831 to 1851 were 92,329 dead in the hospital and only 3,336 killed in action. In 1872, The Algerian population stood at about 2.9 million. French policy was predicated on "civilising" the country. The slave trade and piracy in Algeria ceased following the French conquest. The conquest of Algeria by the French took some time and resulted in considerable bloodshed. A combination of violence and disease epidemics caused the indigenous Algerian population to decline by nearly one-third from 1830 to 1872. On 17 September 1860, Napoleon III declared "Our first duty is to take care of the happiness of the three million Arabs, whom the fate of arms has brought under our domination." During this time, only Kabylia resisted, the Kabylians were not colonised until after the Mokrani Revolt in 1871. Alexis de Tocqueville wrote and never completed an unpublished essay outlining his ideas for how to transform Algeria from an occupied tributary state to a colonial regime, wherein he advocated for a mixed system of "total domination and total colonisation" whereby French military would wage total war against civilian populations while a colonial administration would provide rule of law and property rights to settlers within French occupied cities. From 1848 until independence, France administered the whole Mediterranean region of Algeria as an integral part and département of the nation. One of France's longest-held overseas territories, Algeria became a destination for hundreds of thousands of European immigrants, who became known as colons and later, as Pied-Noirs. Between 1825 and 1847, 50,000 French people emigrated to Algeria. These settlers benefited from the French government's confiscation of communal land from tribal peoples, and the application of modern agricultural techniques that increased the amount of arable land. Many Europeans settled in Oran and Algiers, and by the early 20th century they formed a majority of the population in both cities. During the late 19th and early 20th century, the European share was almost a fifth of the population. The French government aimed at making Algeria an assimilated part of France, and this included substantial educational investments especially after 1900. The indigenous cultural and religious resistance heavily opposed this tendency, but in contrast to the other colonised countries' path in central Asia and Caucasus, Algeria kept its individual skills and a relatively human-capital intensive agriculture. During the Second World War, Algeria came under Vichy control before being liberated by the Allies in Operation Torch, which saw the first large-scale deployment of American troops in the North African campaign. Gradually, dissatisfaction among the Muslim population, which lacked political and economic status under the colonial system, gave rise to demands for greater political autonomy and eventually independence from France. In May 1945, the uprising against the occupying French forces was suppressed through what is now known as the Sétif and Guelma massacre. Tensions between the two population groups came to a head in 1954, when the first violent events of what was later called the Algerian War began after the publication of the Declaration of 1 November 1954. Historians have estimated that between 30,000 and 150,000 Harkis and their dependents were killed by the National Liberation Front (FLN) or by lynch mobs in Algeria. The FLN used hit and run attacks in Algeria and France as part of its war, and the French conducted severe reprisals. In addition, the French destroyed over 8,000 villages and relocated over 2 million Algerians to concentration camps. The war led to the death of hundreds of thousands of Algerians and hundreds of thousands of injuries. Historians, like Alistair Horne and Raymond Aron, state that the actual number of Algerian Muslim war dead was far greater than the original FLN and official French estimates but was less than the 1 million deaths claimed by the Algerian government after independence. Horne estimated Algerian casualties during the span of eight years to be around 700,000. The war uprooted more than 2 million Algerians. The war against French rule concluded in 1962, when Algeria gained complete independence following the March 1962 Evian agreements and the July 1962 self-determination referendum. === The first three decades of independence (1962–1991) === The number of European Pied-Noirs who fled Algeria totaled more than 900,000 between 1962 and 1964. The exodus to mainland France accelerated after the Oran massacre of 1962, in which hundreds of militants entered European sections of the city and began attacking civilians. Algeria's first president was the Front de Libération Nationale (FLN) leader Ahmed Ben Bella. Morocco's claim to portions of western Algeria led to the Sand War in 1963. Ben Bella was overthrown in 1965 by Houari Boumédiène, his former ally and defence minister. Under Ben Bella, the government had become increasingly socialist and authoritarian; Boumédienne continued this trend. However, he relied much more on the army for his support, and reduced the sole legal party to a symbolic role. He collectivised agriculture and launched a massive industrialisation drive. Oil extraction facilities were nationalised. This was especially beneficial to the leadership after the international 1973 oil crisis. Boumédienne's successor, Chadli Bendjedid, introduced some liberal economic reforms. He promoted a policy of Arabisation in Algerian society and public life. Teachers of Arabic, brought in from other Muslim countries, spread conventional Islamic thought in schools and sowed the seeds of a return to Orthodox Islam. The Algerian economy became increasingly dependent on oil, leading to hardship when the price collapsed during the 1980s oil glut. Economic recession caused by the crash in world oil prices resulted in Algerian social unrest during the 1980s; by the end of the decade, Bendjedid introduced a multi-party system. Political parties developed, such as the Islamic Salvation Front (FIS), a broad coalition of Muslim groups. === Civil War (1991–2002) and aftermath === In December 1991 the Islamic Salvation Front dominated the first of two rounds of legislative elections. Fearing the election of an Islamist government, the authorities intervened on 11 January 1992, cancelling the elections. Bendjedid resigned and a High Council of State was installed to act as the Presidency. It banned the FIS, triggering a civil insurgency between the Front's armed wing, the Armed Islamic Group, and the national armed forces, in which more than 100,000 people are thought to have died. The Islamist militants conducted a violent campaign of civilian massacres. At several points in the conflict, the situation in Algeria became a point of international concern, most notably during the crisis surrounding Air France Flight 8969, a hijacking perpetrated by the Armed Islamic Group. The Armed Islamic Group declared a ceasefire in October 1997. Algeria held elections in 1999, considered biased by international observers and most opposition groups which were won by President Abdelaziz Bouteflika. He worked to restore political stability to the country and announced a "Civil Concord" initiative, approved in a referendum, under which many political prisoners were pardoned, and several thousand members of armed groups were granted exemption from prosecution under a limited amnesty, in force until 13 January 2000. The AIS disbanded and levels of insurgent violence fell rapidly. The Groupe Salafiste pour la Prédication et le Combat (GSPC), a splinter group of the Armed Islamic Group, continued a terrorist campaign against the Government. Bouteflika was re-elected in the April 2004 presidential election after campaigning on a programme of national reconciliation. The programme comprised economic, institutional, political and social reform to modernise the country, raise living standards, and tackle the causes of alienation. It also included a second amnesty initiative, the Charter for Peace and National Reconciliation, which was approved in a referendum in September 2005. It offered amnesty to most guerrillas and Government security forces. In November 2008, the Algerian Constitution was amended following a vote in Parliament, removing the two-term limit on Presidential incumbents. This change enabled Bouteflika to stand for re-election in the 2009 presidential elections, and he was re-elected in April 2009. During his election campaign and following his re-election, Bouteflika promised to extend the programme of national reconciliation and a $150-billion spending programme to create three million new jobs, the construction of one million new housing units, and to continue public sector and infrastructure modernisation programmes. A continuing series of protests throughout the country started on 28 December 2010, inspired by similar protests across the Middle East and North Africa. On 24 February 2011, the government lifted Algeria's 19-year-old state of emergency. The government enacted legislation dealing with political parties, the electoral code, and the representation of women in elected bodies. In April 2011, Bouteflika promised further constitutional and political reform. However, elections are routinely criticised by opposition groups as unfair and international human rights groups say that media censorship and harassment of political opponents continue. On 2 April 2019, Bouteflika resigned from the presidency after mass protests against his candidacy for a fifth term in office. In December 2019, Abdelmadjid Tebboune became Algeria's president, after winning the first round of the presidential election with a record abstention rate – the highest of all presidential elections since Algeria's democracy in 1989. Tebboune is accused of being close to the military and being loyal to the deposed president. Tebboune rejects these accusations, claiming to be the victim of a witch hunt. He also reminds his detractors that he was expelled from the Government in August 2017 at the instigation of oligarchs languishing in prison. In September 2024, President Tebboune won a second term with a landslide 84.3 percent of the vote, although his opponents called the results fraud. == Geography == Since the 2011 breakup of Sudan, and the creation of South Sudan, Algeria has been the largest country in Africa, and the Mediterranean Basin. Its southern part includes a significant portion of the Sahara. To the north, the Tell Atlas forms with the Saharan Atlas, further south, two parallel sets of reliefs in approaching eastbound, and between which are inserted vast plains and highlands. Both Atlas tend to merge in eastern Algeria. The vast mountain ranges of Aures and Nememcha occupy the entire northeastern Algeria and are delineated by the Tunisian border. The highest point is Mount Tahat (3,003 metres or 9,852 feet). Algeria lies mostly between latitudes 19° and 37°N (a small area is north of 37°N and south of 19°N), and longitudes 9°W and 12°E. Most of the coastal area is hilly, sometimes even mountainous, and there are a few natural harbours. The area from the coast to the Tell Atlas is fertile. South of the Tell Atlas is a steppe landscape ending with the Saharan Atlas; farther south, there is the Sahara desert. The Hoggar Mountains (Arabic: جبال هقار), also known as the Hoggar, are a highland region in central Sahara, southern Algeria. They are located about 1,500 km (932 mi) south of the capital, Algiers, and just east of Tamanghasset. Algiers, Oran, Constantine, and Annaba are Algeria's main cities. === Climate and hydrology === In this region, midday desert temperatures can be hot year round. After sunset, however, the clear, dry air permits rapid loss of heat, and the nights are cool to chilly. Enormous daily ranges in temperature are recorded. Rainfall is fairly plentiful along the coastal part of the Tell Atlas, ranging from 400 to 670 mm (15.7 to 26.4 in) annually, the amount of precipitation increasing from west to east. Precipitation is heaviest in the northern part of eastern Algeria, where it reaches as much as 1,000 mm (39.4 in) in some years. Farther inland, the rainfall is less plentiful. Algeria also has ergs, or sand dunes, between mountains. Among these, in the summer time when winds are heavy and gusty, temperatures can go up to 43.3 °C (110 °F). === Fauna and flora === The varied vegetation of Algeria includes coastal, mountainous and grassy desert-like regions which all support a wide range of wildlife. In Algeria forest cover is around 1% of the total land area, equivalent to 1,949,000 hectares (ha) of forest in 2020, up from 1,667,000 hectares (ha) in 1990. In 2020, naturally regenerating forest covered 1,439,000 hectares (ha) and planted forest covered 510,000 hectares (ha). Of the naturally regenerating forest 0% was reported to be primary forest (consisting of native tree species with no clearly visible indications of human activity) and around 6% of the forest area was found within protected areas. For the year 2015, 80% of the forest area was reported to be under public ownership, 18% private ownership and 2% with ownership listed as other or unknown. Many of the creatures constituting the Algerian wildlife live in close proximity to civilisation. The most commonly seen animals include the wild boars, jackals, and gazelles, although it is not uncommon to spot fennecs (foxes), and jerboas. Algeria also has a small African leopard and Saharan cheetah population, but these are seldom seen. A species of deer, the Barbary stag, inhabits the dense humid forests in the north-eastern areas. The fennec fox is the national animal of Algeria. A variety of bird species makes the country an attraction for bird watchers. The forests are inhabited by boars and jackals. Barbary macaques are the sole native monkey. Snakes, monitor lizards, and numerous other reptiles can be found living among an array of rodents throughout the semi arid regions of Algeria. Many animals are now extinct, including the Barbary lions, Atlas bears and crocodiles. In the north, some of the native flora includes Macchia scrub, olive trees, oaks, cedars and other conifers. The mountain regions contain large forests of evergreens (Aleppo pine, juniper, and evergreen oak) and some deciduous trees. Fig, eucalyptus, agave, and various palm trees grow in the warmer areas. The grape vine is indigenous to the coast. In the Sahara region, some oases have palm trees. Acacias with wild olives are the predominant flora in the remainder of the Sahara. Algeria had a 2018 Forest Landscape Integrity Index mean score of 5.22/10, ranking it 106th globally out of 172 countries. Camels are used extensively; the desert also abounds with venomous and nonvenomous snakes, scorpions, and numerous insects. == Government and politics == Algeria's government has been described as authoritarian, and elected politicians have relatively little sway over affairs in the country. Instead, a group of unelected civilian and military "décideurs" ("deciders") known as "le pouvoir" ("the power") exercising de facto rule over the country, even deciding who should be president. The most powerful man might have been Mohamed Mediène, the head of military intelligence, before he was brought down during the 2019 protests. In recent years, many of these generals have died, retired, or been imprisoned. After the death of General Larbi Belkheir, previous president Bouteflika put loyalists in key posts, notably at Sonatrach, and secured constitutional amendments that made him re-electable indefinitely, until he was brought down in 2019 during protests. The head of state is the President of Algeria, who is elected for a five-year term. The president is limited to two five-year terms. The most recent presidential election was planned to be in April 2019, but widespread protests erupted on 22 February against the president's decision to participate in the election, which resulted in President Bouteflika announcing his resignation on 3 April. Abdelmadjid Tebboune, an independent candidate, was elected as president after the election eventually took place on 12 December 2019. Protestors refused to recognise Tebboune as president, citing demands for comprehensive reform of the political system. Algeria has universal suffrage at 18 years of age. The President is the head of the army, the Council of Ministers and the High Security Council. He appoints the Prime Minister who is also the head of government. The Algerian parliament is bicameral; the lower house, the People's National Assembly, has 462 members who are directly elected for five-year terms, while the upper house, the Council of the Nation, has 144 members serving six-year terms, of which 96 members are chosen by local assemblies and 48 are appointed by the president. According to the constitution, no political association may be formed if it is "based on differences in religion, language, race, gender, profession, or region". In addition, political campaigns must be exempt from the aforementioned subjects. Parliamentary elections were last held in June 2021. In the elections, the FLN lost 66 of its seats, but remained the largest party with 98 seats. Other parties included the Movement of the Society for Peace which won 65 seats, the National Rally for Democracy which won 58 seats, the Future Front which won 48 seats, and the National Construction Movement which won 39 seats. === Foreign relations === Algeria is included in the European Union's European Neighbourhood Policy (ENP) which aims at bringing the EU and its neighbours closer. Giving incentives and rewarding best performers, as well as offering funds in a faster and more flexible manner, are the two main principles underlying the European Neighbourhood Instrument (ENI) that came into force in 2014. It has a budget of €15.4 billion and provides the bulk of funding through a number of programmes. In 2009, the French government agreed to compensate victims of nuclear tests in Algeria. Defence Minister Hervé Morin stated that "It's time for our country to be at peace with itself, at peace thanks to a system of compensation and reparations", when presenting the draft law on the payouts. Algerian officials and activists believe that this is a good first step and hope that this move would encourage broader reparation. Tensions between Algeria and Morocco in relation to the Western Sahara have been an obstacle to tightening the Arab Maghreb Union, nominally established in 1989, but which has carried little practical weight. On 24 August 2021, Algeria announced the break of diplomatic relations with Morocco. === Military === The military of Algeria consists of the People's National Army (ANP), the Algerian National Navy (MRA), and the Algerian Air Force (QJJ), plus the Territorial Air Defence Forces. It is the direct successor of the National Liberation Army (Armée de Libération Nationale or ALN), the armed wing of the nationalist National Liberation Front which fought French colonial occupation during the Algerian War of Independence (1954–62). Total military personnel include 147,000 active, 150,000 reserve, and 187,000 paramilitary staff (2008 estimate). Service in the military is compulsory for men aged 19–30, for a total of 12 months. The military expenditure was 4.3% of the gross domestic product (GDP) in 2012. Algeria has the second-largest military in North Africa with the largest defence budget in Africa ($10 billion). Most of Algeria's weapons are imported from Russia, with whom they are a close ally. In 2007, the Algerian Air Force signed a deal with Russia to purchase 49 MiG-29SMT and 6 MiG-29UBT at an estimated cost of $1.9 billion. Russia is also building two 636-type diesel submarines for Algeria. Algeria is the 90th most peaceful country in the world, according to the 2024 Global Peace Index. === Human rights === Algeria has been categorised by the US government funded Freedom House as "not free" since it began publishing such ratings in 1972, with the exception of 1989, 1990, and 1991, when the country was labelled "partly free". In December 2016, the Euro-Mediterranean Human Rights Monitor issued a report regarding violation of media freedom in Algeria. It clarified that the Algerian government imposed restrictions on freedom of the press; expression; and right to peaceful demonstration, protest and assembly as well as intensified censorship of the media and websites. Due to the fact that the journalists and activists criticise the ruling government, some media organisations' licenses are cancelled. Independent and autonomous trade unions face routine harassment from the government, with many leaders imprisoned and protests suppressed. In 2016, a number of unions, many of which were involved in the 2010–2012 Algerian Protests, have been deregistered by the government. Homosexuality is illegal in Algeria. Public homosexual behavior is punishable by up to two years in prison. Despite this, about 26% of Algerians think that homosexuality should be accepted, according to the survey conducted by the BBC News Arabic-Arab Barometer in 2019. Algeria showed the highest LGBT acceptance compared to other Arab countries where the survey was conducted. Human Rights Watch has accused the Algerian authorities of using the COVID-19 pandemic as an excuse to prevent pro-democracy movements and protests in the country, leading to the arrest of youths as part of social distancing. === Administrative divisions === Algeria is divided into 58 provinces (wilayas), 553 districts (daïras) and 1,541 municipalities (baladiyahs). Each province, district, and municipality is named after its seat, which is usually the largest city. The administrative divisions have changed several times since independence. When introducing new provinces, the numbers of old provinces are kept, hence the non-alphabetical order. With their official numbers, currently (since 1983) they are: == Economy == Algeria's currency is the dinar (DZD). The economy remains dominated by the state, a legacy of the country's socialist post-independence development model. In June 2024 The World Bank's 2024 report marks a turning point for Algeria, which joins the select club of upper-middle-income countries. This economic rise, the result of an ambitious development strategy, places the country in the same category as emerging powers such as China, Brazil and Turkey In recent years, the Algerian government has halted the privatisation of state-owned industries and imposed restrictions on imports and foreign involvement in its economy. These restrictions are just starting to be lifted off recently although questions about Algeria's slowly-diversifying economy remain. Algeria has struggled to develop industries outside hydrocarbons in part because of high costs and an inert state bureaucracy. The government's efforts to diversify the economy by attracting foreign and domestic investment outside the energy sector have done little to reduce high youth unemployment rates or to address housing shortages. The country is facing a number of short-term and medium-term problems, including the need to diversify the economy, strengthen political, economic and financial reforms, improve the business climate and reduce inequalities among regions. A wave of economic protests in February and March 2011 prompted the Algerian government to offer more than $23 billion in public grants and retroactive salary and benefit increases. Public spending has increased by 27% annually during the past five years. The 2010–14 public-investment programme will cost US$286 billion, 40% of which will go to human development. Thanks to strong hydrocarbon revenues, Algeria has a cushion of $173 billion in foreign currency reserves and a large hydrocarbon stabilisation fund. In addition, Algeria's external debt is extremely low at about 2% of GDP. The economy remains very dependent on hydrocarbon wealth, and, despite high foreign exchange reserves (US$178 billion, equivalent to three years of imports), current expenditure growth makes Algeria's budget more vulnerable to the risk of prolonged lower hydrocarbon revenues. Algeria has not joined the WTO, despite several years of negotiations but is a member of the Greater Arab Free Trade Area, the African Continental Free Trade Area, and has an association agreement with the European Union. Turkish direct investments have accelerated in Algeria, with total value reaching $5 billion. As of 2022, the number of Turkish companies present in Algeria has reached 1,400. In 2020, despite the pandemic, more than 130 Turkish companies were created in Algeria. === Oil and natural resources === Algeria, whose economy is reliant on petroleum, has been an OPEC member since 1969. Its crude oil production stands at around 1.1 million barrels/day, but it is also a major gas producer and exporter, with important links to Europe. Hydrocarbons have long been the backbone of the economy, accounting for roughly 60% of budget revenues, 30% of GDP, and 87.7% of export earnings. Algeria has the 10th-largest reserves of natural gas in the world and is the sixth-largest gas exporter. The U.S. Energy Information Administration reported that in 2005, Algeria had 4.5 trillion cubic metres (160×10^12 cu ft) of proven natural gas reserves. It also ranks 16th in oil reserves. Non-hydrocarbon growth for 2011 was projected at 5%. To cope with social demands, the authorities raised expenditure, especially on basic food support, employment creation, support for SMEs, and higher salaries. High hydrocarbon prices have improved the current account and the already large international reserves position. Income from oil and gas rose in 2011 as a result of continuing high oil prices, though the trend in production volume is downward. Production from the oil and gas sector in terms of volume continues to decline, dropping from 43.2 million tonnes to 32 million tonnes between 2007 and 2011. Nevertheless, the sector accounted for 98% of the total volume of exports in 2011, against 48% in 1962, and 70% of budgetary receipts, or US$71.4 billion. The Algerian national oil company is Sonatrach, which plays a key role in all aspects of the oil and natural gas sectors in Algeria. All foreign operators must work in partnership with Sonatrach, which usually has majority ownership in production-sharing agreements. Access to biocapacity in Algeria is lower than world average. In 2016, Algeria had 0.53 global hectares of biocapacity per person within its territory, much less than the world average of 1.6 global hectares per person. In 2016, Algeria used 2.4 global hectares of biocapacity per person – their ecological footprint of consumption. This means they use just under 4.5 times as much biocapacity as Algeria contains. As a result, Algeria is running a biocapacity deficit. In April 2022, diplomats from Italy and Spain held talks after Rome's move to secure large volume of Algerian gas stoked concerns in Madrid. Under the deal between Algeria's Sonatrach and Italy's Eni, Algeria will send an additional 9 billion cubic metres of gas to Italy by next year and in 2024. === Research and alternative energy sources === Algeria has invested an estimated 100 billion dinars towards developing research facilities and paying researchers. This development programme is meant to advance alternative energy production, especially solar and wind power. Algeria is estimated to have the largest solar energy potential in the Mediterranean, so the government has funded the creation of a solar science park in Hassi R'Mel. Currently, Algeria has 20,000 research professors at various universities and over 780 research labs, with state-set goals to expand to 1,000. Besides solar energy, areas of research in Algeria include space and satellite telecommunications, nuclear power and medical research. === Labour market === The overall rate of unemployment was 11.8% in 2023. The government strengthened in 2011 the job programmes introduced in 1988, in particular in the framework of the programme to aid those seeking work (Dispositif d'Aide à l'Insertion Professionnelle). Despite a decline in total unemployment, youth and women unemployment is high. === Tourism === The development of the tourism sector in Algeria had previously been hampered by a lack of facilities, but since 2004 a broad tourism development strategy has been implemented resulting in many hotels of a high modern standard being built. There are several UNESCO World Heritage Sites in Algeria which includes Al Qal'a of Beni Hammad, the first capital of the Hammadid empire; Tipasa, a Phoenician and later Roman town;Djémila and Timgad, both Roman ruins; M'Zab Valley, a limestone valley containing a large urbanised oasis; and the Casbah of Algiers, an important citadel. The only natural World Heritage Site in Algeria is the Tassili n'Ajjer, a mountain range. === Transport === Two trans-African automobile routes pass through Algeria: Cairo-Dakar Highway Algiers-Lagos Highway The Algerian road network is the densest in Africa; its length is estimated at 180,000 km (110,000 mi) of highways, with more than 3,756 structures and a paving rate of 85%. This network will be complemented by the East-West Highway, a major infrastructure project currently under construction. It is a three-way, 1,216-kilometre-long (756 mi) highway, linking Annaba in the extreme east to the Tlemcen in the far west. Algeria is also crossed by the Trans-Sahara Highway, which is now completely paved. This road is supported by the Algerian government to increase trade between the six countries crossed: Algeria, Mali, Niger, Nigeria, Chad, and Tunisia. == Demographics == Algeria has a population of an estimated 45.6 million, of which the majority, 75% to 85% are ethnically Arab. At the outset of the 20th century, its population was approximately 4 million. About 90% of Algerians live in the northern, coastal area; the inhabitants of the Sahara desert are mainly concentrated in oases, although some 1.5 million remain nomadic or partly nomadic. 28.1% of Algerians are under the age of 15. Between 90,000 and 165,000 Sahrawis from Western Sahara live in the Sahrawi refugee camps, in the western Algerian Sahara desert. There are also more than 4,000 Palestinian refugees, who are well integrated and have not asked for assistance from the United Nations High Commissioner for Refugees (UNHCR). In 2009, 35,000 Chinese migrant workers lived in Algeria. The largest concentration of Algerian migrants outside Algeria is in France, which has reportedly over 1.7 million Algerians of up to the second generation. There are also many foreign communities in Algeria, though these do not make up a significant portion of the population. === Ethnic groups === Arabs and indigenous Berbers as well as Phoenicians, Romans, Vandals, Byzantine Greeks, Turks, various Sub-Saharan Africans, and French have contributed to the history and culture of Algeria. Descendants of Andalusi refugees are also present in the population of Algiers and other cities. Moreover, Spanish was spoken by these Aragonese and Castillian Morisco descendants deep into the 18th century, and even Catalan was spoken at the same time by Catalan Morisco descendants in the small town of Grish El-Oued. Centuries of Arab migrations to the Maghreb since the seventh century shifted the demographic scope in Algeria. Estimates vary based on different sources. The majority of the population of Algeria is ethnically Arab, constituting between 75% and 80% to 85% of the population. Berbers who make up between 15% and 20% to 24% of the population are divided into many groups with varying languages. The largest of these are the Kabyles, who live in the Kabylie region east of Algiers, the Chaoui of Northeast Algeria, the Tuaregs in the southern desert and the Shenwa people of North Algeria. During the colonial period, there was a large (10% in 1960) European population who became known as Pied-Noirs. They were primarily of French, Spanish and Italian origin. Almost all of this population left during the war of independence or immediately after its end. === Languages === Modern Standard Arabic and Berber are the official languages. Algerian Arabic (Darja) is the language used by the majority of the population. Colloquial Algerian Arabic has some Berber loanwords which represent 8% to 9% of its vocabulary. Berber has been recognised as a "national language" by the constitutional amendment of 8 May 2002. Kabyle, the predominant Berber language, is taught and is partially co-official (with a few restrictions) in parts of Kabylie. Kabyle has a significant Arabic, French, Latin, Greek, Phoenician and Punic substratum, and Arabic loanwords represent 35% of the total Kabyle vocabulary. In February 2016, the Algerian constitution passed a resolution that made Berber an official language alongside Arabic. Algeria emerged as a bilingual state after 1962. Colloquial Algerian Arabic is spoken by about 83% of the population and Berber by 27%. Although French has no official status in Algeria, it has one of the largest Francophone populations in the world, and French is widely used in government, media (newspapers, radio, local television), and both the education system (from primary school onwards) and academia due to Algeria's colonial history. It can be regarded as a lingua franca of Algeria. In 2008, 11.2 million Algerians could read and write in French. In 2013, it was estimated that 60% of the population could speak or understand French. In 2022, it was estimated that 33% of the population was Francophone. The use of English in Algeria, though limited in comparison to the previously mentioned languages, has increased due to globalisation. In 2022 it was announced that English would be taught in elementary schools. === Religion === Islam is the predominant religion in Algeria, with its adherents, mostly Sunnis, accounting for 99% of the population according to a 2021 CIA World Factbook estimate, and 97.9% according to Pew Research in 2020. There are about 290,000 Ibadis in the M'zab Valley in the region of Ghardaia. Prior to independence, Algeria was home to more than 1.3 million Christians (mostly of European ancestry). Most of the Christian settlers left to France after the country's independence. Today, estimates of the Christian population range from 100,000 to 200,000. Algerian citizens who are Christians predominantly belong to Protestant denominations, which have seen increased pressure from the government in recent years including many forced closures. According to the Arab Barometer in 2018–2019, the vast majority of Algerians (99.1%) continue to identify as Muslim. The June 2019 Arab Barometer-BBC News report found that the percentage of Algerians identifying as non-religious has grown from around 8% in 2013 to around 15% in 2018. The Arab Barometer December 2019, found that the growth in the percentage of Algerians identifying as non-religious is largely driven by young Algerians, with roughly 25% describing themselves as non-religious. However, the 2021 Arab Barometer report found that those who said they were not religious among Algerians has decreased, with just 2.6% identifying as non-religious. In that same report, 69.5% of Algerians identified as religious and another 27.8% identifying as somewhat religious. Algeria has given the Muslim world a number of prominent thinkers, including Emir Abdelkader, Abdelhamid Ben Badis, Mouloud Kacem Naît Belkacem, Malek Bennabi and Mohamed Arkoun. === Health === In 2018, Algeria had the highest numbers of physicians in the Maghreb region (1.72 per 1,000 people), nurses (2.23 per 1,000 people), and dentists (0.31 per 1,000 people). Access to "improved water sources" was around 97.4% of the population in urban areas and 98.7% of the population in the rural areas. Some 99% of Algerians living in urban areas, and around 93.4% of those living in rural areas, had access to "improved sanitation". According to the World Bank, Algeria is making progress towards its goal of "reducing by half the number of people without sustainable access to improved drinking water and basic sanitation by 2015". Given Algeria's young population, policy favours preventive health care and clinics over hospitals. In keeping with this policy, the government maintains an immunisation programme. However, poor sanitation and unclean water still cause tuberculosis, hepatitis, measles, typhoid fever, cholera and dysentery. The poor generally receive healthcare free of charge. Health records have been maintained in Algeria since 1882 and began adding Muslims living in the south to their vital record database in 1905 during French rule. === Education === Since the 1970s, in a centralised system that was designed to significantly reduce the rate of illiteracy, the Algerian government introduced a decree by which school attendance became compulsory for all children aged between 6 and 15 years who have the ability to track their learning through the 20 facilities built since independence, now the literacy rate is around 92.6%. Since 1972, Arabic is used as the language of instruction during the first nine years of schooling. From the third year, French is taught and it is also the language of instruction for science classes. The students can also learn English, Italian, Spanish and German. In 2008, new programmes at the elementary appeared, therefore the compulsory schooling does not start at the age of six anymore, but at the age of five. Apart from the 122 private schools, the Universities of the State are free of charge. After nine years of primary school, students can go to a high school or to an educational institution. The school offers two programmes: general or technical. At the end of the third year of secondary school, students pass the exam of the baccalaureate, which allows once it is successful to pursue graduate studies in universities and institutes. Education is officially compulsory for children between the ages of six and 15. In 2008, the illiteracy rate for people over 10 was 22.3%, 15.6% for men and 29.0% for women. The province with the lowest rate of illiteracy was Algiers Province at 11.6%, while the province with the highest rate was Djelfa Province at 35.5%. Algeria has 26 universities and 67 institutions of higher education, which must accommodate a million Algerians and 80,000 foreign students in 2008. The University of Algiers, founded in 1879, is the oldest, it offers education in various disciplines (law, medicine, science and letters). Twenty-five of these universities and almost all of the institutions of higher education were founded after the independence of the country. Even if some of them offer instruction in Arabic like areas of law and the economy, most of the other sectors such as science and medicine continue to be provided in French and English. Among the most important universities, there are the University of Sciences and Technology Houari Boumediene, the University of Mentouri Constantine, and University of Oran Es-Senia. The University of Abou Bekr Belkaïd in Tlemcen and University of Batna Hadj Lakhdar occupy the 26th and 45th row in Africa. Algeria was ranked 115th in the Global Innovation Index in 2024. == Culture == Modern Algerian literature, split between Arabic, Tamazight and French, has been strongly influenced by the country's recent history. Famous novelists of the 20th century include Mohammed Dib, Albert Camus, Kateb Yacine and Ahlam Mosteghanemi while Assia Djebar is widely translated. Among the important novelists of the 1980s were Rachid Mimouni, later vice-president of Amnesty International, and Tahar Djaout, murdered by an Islamist group in 1993 for his secularist views. Malek Bennabi and Frantz Fanon are noted for their thoughts on decolonisation; Augustine of Hippo was born in Tagaste (modern-day Souk Ahras); and Ibn Khaldun, though born in Tunis, wrote the Muqaddima while staying in Algeria. The works of the Sanusi family in pre-colonial times, and of Emir Abdelkader and Sheikh Ben Badis in colonial times, are widely noted. The Latin author Apuleius was born in Madaurus (Mdaourouch), in what later became Algeria. Contemporary Algerian cinema is varied in terms of genre, exploring a wider range of themes and issues. There has been a transition from cinema which focused on the war of independence to films more concerned with the everyday lives of Algerians. === Media === === Art === Algerian painters, like Mohammed Racim and Baya, attempted to revive the prestigious Algerian past prior to French colonisation, at the same time that they have contributed to the preservation of the authentic values of Algeria. In this line, Mohamed Temam, Abdelkhader Houamel have also returned through this art, scenes from the history of the country, the habits and customs of the past and the country life. Other new artistic currents including the one of M'hamed Issiakhem, Mohammed Khadda and Bachir Yelles, appeared on the scene of Algerian painting, abandoning figurative classical painting to find new pictorial ways, to adapt Algerian paintings to the new realities of the country through its struggle and its aspirations. Mohammed Khadda and M'hamed Issiakhem have been notable in recent years. === Literature === The historic roots of Algerian literature go back to the Numidian and Roman African era, when Apuleius wrote The Golden Ass, the only Latin novel to survive in its entirety. This period had also known Augustine of Hippo, Nonius Marcellus and Martianus Capella, among many others. The Middle Ages have known many Arabic writers who revolutionised the Arab world literature, with authors like Ahmad al-Buni, Ibn Manzur and Ibn Khaldoun, who wrote the Muqaddimah while staying in Algeria, and many others. Albert Camus was an Algerian-born French Pied-Noir author. In 1957, he was awarded the Nobel Prize in Literature. Today Algeria contains, in its literary landscape, big names having not only marked the Algerian literature, but also the universal literary heritage in Arabic and French. As a first step, Algerian literature was marked by works whose main concern was the assertion of the Algerian national entity, there is the publication of novels as the Algerian trilogy of Mohammed Dib, or even Nedjma of Kateb Yacine novel which is often regarded as a monumental and major work. Other known writers will contribute to the emergence of Algerian literature whom include Mouloud Feraoun, Malek Bennabi, Malek Haddad, Moufdi Zakaria, Abdelhamid Ben Badis, Mohamed Laïd Al-Khalifa, Mouloud Mammeri, Frantz Fanon, and Assia Djebar. In the aftermath of the independence, several new authors emerged on the Algerian literary scene, they will attempt through their works to expose a number of social problems, among them there are Rachid Boudjedra, Rachid Mimouni, Leila Sebbar, Tahar Djaout and Tahir Wattar. Currently, a part of Algerian writers tends to be defined in a literature of shocking expression, due to the terrorism that occurred during the 1990s, the other party is defined in a different style of literature who staged an individualistic conception of the human adventure. Among the most noted recent works, there is the writer, the swallows of Kabul and the attack of Yasmina Khadra, the oath of barbarians of Boualem Sansal, memory of the flesh of Ahlam Mosteghanemi and the last novel by Assia Djebar nowhere in my father's House. === Cinema === The Algerian state's interest in film-industry activities can be seen in the annual budget of DZD 200 million (EUR 1.3 million) allocated to production, specific measures and an ambitious programme plan implemented by the Ministry of Culture to promote national production, renovate the cinema stock and remedy the weak links in distribution and exploitation. The financial support provided by the state, through the Fund for the Development of the Arts, Techniques and the Film Industry (FDATIC) and the Algerian Agency for Cultural Influence (AARC), plays a key role in the promotion of national production. Between 2007 and 2013, FDATIC subsidised 98 films (feature films, documentaries and short films). In mid-2013, AARC had already supported a total of 78 films, including 42 feature films, 6 short films and 30 documentaries. According to the European Audiovisual Observatory's LUMIERE database, 41 Algerian films were distributed in Europe between 1996 and 2013; 21 films in this repertoire were Algerian-French co-productions. Days of Glory (2006) and Outside the Law (2010) recorded the highest number of admissions in the European Union, 3,172,612 and 474,722, respectively. Algeria won the Palme d'Or for Chronicle of the Years of Fire (1975), two Oscars for Z (1969), and other awards for the Italian-Algerian movie The Battle of Algiers. === Cuisine === Algerian cuisine is rich and diverse as a result of interactions and exchanges with other cultures and nations over the centuries. It is based on both land and sea products. Conquests or demographic movement towards the Algerian territory were two of the main factors of exchanges between the different peoples and cultures. The Algerian cuisine is a mix of Arab, Berber, Turkish and French roots. Algerian cuisine offers a variety of dishes depending on the region and the season, but vegetables and cereals remain at its core. Most of the Algerian dishes are centered around bread, meats (lamb, beef or poultry), olive oil, vegetables, and fresh herbs. Vegetables are often used for salads, soups, tajines, couscous, and sauce-based dishes. Of all the Algerian traditional dishes available, the most famous one is couscous, recognized as a national dish. === Sports === Various games have existed in Algeria since antiquity. In the Aures, people played several games such as El Kherba or El khergueba (chess variant). Playing cards, checkers and chess games are part of Algerian culture. Racing (fantasia) and rifle shooting are part of cultural recreation of the Algerians. Football is the most popular sport in the country. The Algerian national football team, known as the Desert Foxes, has a strong fan base and has achieved success both domestically and internationally. Algeria has a long history in other sports such as athletics, boxing, volleyball, handball and the study of martial arts. Algerian athletes have competed in the Olympic Games and have won medals in various events. Many sports clubs and organisations exist in Algeria to promote and develop sports among young people. The Ministry of Youth and Sports in Algeria manages sport-related activities. == See also == Index of Algeria-related articles Outline of Algeria == Explanatory notes == == Citations == == General bibliography == == External links == Key Development Forecasts for Algeria from International Futures === Government === Public Services – gateway to government sites El Mouradia Palace – official website of the president of Algeria Statistics – official website of National Office of Statistics === History === "History" – Algerian history at Ministry of Foreign Affairs === Tourism === Visit Algeria Archived 6 February 2023 at the Wayback Machine – Algeria's official tourism portal === Maps === Wikimedia Atlas of Algeria Geographic data related to Algeria at OpenStreetMap
Wikipedia:Algebra & Number Theory#0	Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. == History == === Diophantus === The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively: A = x + y {\displaystyle A=x+y\ } B = x 2 + y 2 . {\displaystyle B=x^{2}+y^{2}.\ } Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). Diophantus's major work was the Arithmetica, of which only a portion has survived. === Fermat === Fermat's Last Theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. === Gauss === One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular. === Dirichlet === In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers. === Dedekind === Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: "Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983) 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. === Hilbert === David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area. He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi. === Artin === Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem. === Modern theory === Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form. It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, yet no proof; as a result the "astounding" conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands program, a list of important conjectures needing proof or disproof. From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat. == Basic notions == === Failure of unique factorization === An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K. A prime element is an element p of O such that if p divides a product ab, then it divides one of the factors a or b. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. For example, −2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as 6 = 2 ⋅ 3 = ( − 2 ) ⋅ ( − 3 ) . {\displaystyle 6=2\cdot 3=(-2)\cdot (-3).} In general, if u is a unit, meaning a number with a multiplicative inverse in O, and if p is a prime element, then up is also a prime element. Numbers such as p and up are said to be associate. In the integers, the primes p and −p are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers Z[i], the numbers 1 + 2i and −2 + i are associate because the latter is the product of the former by i, but there is no way to single out one as being more canonical than the other. This leads to equations such as 5 = ( 1 + 2 i ) ( 1 − 2 i ) = ( 2 + i ) ( 2 − i ) , {\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),} which prove that in Z[i], it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering. However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An irreducible element x is an element such that if x = yz, then either y or z is a unit. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring Z[√-5]. In this ring, the numbers 3, 2 + √-5 and 2 - √-5 are irreducible. This means that the number 9 has two factorizations into irreducible elements, 9 = 3 2 = ( 2 + − 5 ) ( 2 − − 5 ) . {\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).} This equation shows that 3 divides the product (2 + √-5)(2 - √-5) = 9. If 3 were a prime element, then it would divide 2 + √-5 or 2 - √-5, but it does not, because all elements divisible by 3 are of the form 3a + 3b√-5. Similarly, 2 + √-5 and 2 - √-5 divide the product 32, but neither of these elements divides 3 itself, so neither of them are prime. As there is no sense in which the elements 3, 2 + √-5 and 2 - √-5 can be made equivalent, unique factorization fails in Z[√-5]. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective. === Factorization into prime ideals === If I is an ideal in O, then there is always a factorization I = p 1 e 1 ⋯ p t e t , {\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},} where each p i {\displaystyle {\mathfrak {p}}_{i}} is a prime ideal, and where this expression is unique up to the order of the factors. In particular, this is true if I is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are Dedekind domains. When O is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. In Z[√-5], for instance, the ideal (2, 1 + √-5) is a prime ideal which cannot be generated by a single element. Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field E of K. This extension field is now known as the Hilbert class field. By the principal ideal theorem, every prime ideal of O generates a principal ideal of the ring of integers of E. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in cyclotomic fields. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals. An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals pZ are prime ideals of the ring Z. However, when this ideal is extended to the Gaussian integers to obtain pZ[i], it may or may not be prime. For example, the factorization 2 = (1 + i)(1 − i) implies that 2 Z [ i ] = ( 1 + i ) Z [ i ] ⋅ ( 1 − i ) Z [ i ] = ( ( 1 + i ) Z [ i ] ) 2 ; {\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf {Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf {Z} [i])^{2};} note that because 1 + i = (1 − i) ⋅ i, the ideals generated by 1 + i and 1 − i are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares. It implies that for an odd prime number p, pZ[i] is a prime ideal if p ≡ 3 (mod 4) and is not a prime ideal if p ≡ 1 (mod 4). This, together with the observation that the ideal (1 + i)Z[i] is prime, provides a complete description of the prime ideals in the Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (that is, a Galois extension with abelian Galois group). === Ideal class group === Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a group structure. This is done by generalizing ideals to fractional ideals. A fractional ideal is an additive subgroup J of K which is closed under multiplication by elements of O, meaning that xJ ⊆ J if x ∈ O. All ideals of O are also fractional ideals. If I and J are fractional ideals, then the set IJ of all products of an element in I and an element in J is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal (1) = O, and the inverse of J is a (generalized) ideal quotient: J − 1 = ( O : J ) = { x ∈ K : x J ⊆ O } . {\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.} The principal fractional ideals, meaning the ones of the form Ox where x ∈ K×, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals I and J represent the same element of the ideal class group if and only if there exists an element x ∈ K such that xI = J. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted Cl K, Cl O, or Pic O (with the last notation identifying it with the Picard group in algebraic geometry). The number of elements in the class group is called the class number of K. The class number of Q(√-5) is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as (2, 1 + √-5). The ideal class group has another description in terms of divisors. These are formal objects which represent possible factorizations of numbers. The divisor group Div K is defined to be the free abelian group generated by the prime ideals of O. There is a group homomorphism from K×, the non-zero elements of K up to multiplication, to Div K. Suppose that x ∈ K satisfies ( x ) = p 1 e 1 ⋯ p t e t . {\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.} Then div x is defined to be the divisor div ⁡ x = ∑ i = 1 t e i [ p i ] . {\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].} The kernel of div is the group of units in O, while the cokernel is the ideal class group. In the language of homological algebra, this says that there is an exact sequence of abelian groups (written multiplicatively), 1 → O × → K × → div Div ⁡ K → Cl ⁡ K → 1. {\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.} === Real and complex embeddings === Some number fields, such as Q(√2), can be specified as subfields of the real numbers. Others, such as Q(√−1), cannot. Abstractly, such a specification corresponds to a field homomorphism K → R or K → C. These are called real embeddings and complex embeddings, respectively. A real quadratic field Q(√a), with a ∈ Q, a > 0, and a not a perfect square, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send √a to √a and to −√a, respectively. Dually, an imaginary quadratic field Q(√−a) admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends √−a to √−a, while the other sends it to its complex conjugate, −√−a. Conventionally, the number of real embeddings of K is denoted r1, while the number of conjugate pairs of complex embeddings is denoted r2. The signature of K is the pair (r1, r2). It is a theorem that r1 + 2r2 = d, where d is the degree of K. Considering all embeddings at once determines a function M : K → R r 1 ⊕ C r 2 {\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}} , or equivalently M : K → R r 1 ⊕ R 2 r 2 . {\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.} This is called the Minkowski embedding. The subspace of the codomain fixed by complex conjugation is a real vector space of dimension d called Minkowski space. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of K by an element x ∈ K corresponds to multiplication by a diagonal matrix in the Minkowski embedding. The dot product on Minkowski space corresponds to the trace form ⟨ x , y ⟩ = Tr ⁡ ( x y ) {\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)} . The image of O under the Minkowski embedding is a d-dimensional lattice. If B is a basis for this lattice, then det BTB is the discriminant of O. The discriminant is denoted Δ or D. The covolume of the image of O is \| Δ \| {\displaystyle {\sqrt {\|\Delta \|}}} . === Places === Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on valuations. Consider, for example, the integers. In addition to the usual absolute value function \|·\| : Q → R, there are p-adic absolute value functions \|·\|p : Q → R, defined for each prime number p, which measure divisibility by p. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of Q and the prime numbers. A place of an algebraic number field is an equivalence class of absolute value functions on K. There are two types of places. There is a p {\displaystyle {\mathfrak {p}}} -adic absolute value for each prime ideal p {\displaystyle {\mathfrak {p}}} of O, and, like the p-adic absolute values, it measures divisibility. These are called finite places. The other type of place is specified using a real or complex embedding of K and the standard absolute value function on R or C. These are infinite places. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are r1 real places and r2 complex places. Because places encompass the primes, places are sometimes referred to as primes. When this is done, finite places are called finite primes and infinite places are called infinite primes. If v is a valuation corresponding to an absolute value, then one frequently writes v ∣ ∞ {\displaystyle v\mid \infty } to mean that v is an infinite place and v ∤ ∞ {\displaystyle v\nmid \infty } to mean that it is a finite place. Considering all the places of the field together produces the adele ring of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the Artin reciprocity law. ==== Places at infinity geometrically ==== There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let k = F q {\displaystyle k=\mathbb {F} _{q}} and X / k {\displaystyle X/k} be a smooth, projective, algebraic curve. The function field F = k ( X ) {\displaystyle F=k(X)} has many absolute values, or places, and each corresponds to a point on the curve. If X {\displaystyle X} is the projective completion of an affine curve X ^ ⊂ A n {\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}} then the points in X − X ^ {\displaystyle X-{\hat {X}}} correspond to the places at infinity. Then, the completion of F {\displaystyle F} at one of these points gives an analogue of the p {\displaystyle p} -adics. For example, if X = P 1 {\displaystyle X=\mathbb {P} ^{1}} then its function field is isomorphic to k ( t ) {\displaystyle k(t)} where t {\displaystyle t} is an indeterminant and the field F {\displaystyle F} is the field of fractions of polynomials in t {\displaystyle t} . Then, a place v p {\displaystyle v_{p}} at a point p ∈ X {\displaystyle p\in X} measures the order of vanishing or the order of a pole of a fraction of polynomials p ( x ) / q ( x ) {\displaystyle p(x)/q(x)} at the point p {\displaystyle p} . For example, if p = [ 2 : 1 ] {\displaystyle p=[2:1]} , so on the affine chart x 1 ≠ 0 {\displaystyle x_{1}\neq 0} this corresponds to the point 2 ∈ A 1 {\displaystyle 2\in \mathbb {A} ^{1}} , the valuation v 2 {\displaystyle v_{2}} measures the order of vanishing of p ( x ) {\displaystyle p(x)} minus the order of vanishing of q ( x ) {\displaystyle q(x)} at 2 {\displaystyle 2} . The function field of the completion at the place v 2 {\displaystyle v_{2}} is then k ( ( t − 2 ) ) {\displaystyle k((t-2))} which is the field of power series in the variable t − 2 {\displaystyle t-2} , so an element is of the form a − k ( t − 2 ) − k + ⋯ + a − 1 ( t − 2 ) − 1 + a 0 + a 1 ( t − 2 ) + a 2 ( t − 2 ) 2 + ⋯ = ∑ n = − k ∞ a n ( t − 2 ) n {\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-2)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\end{aligned}}} for some k ∈ N {\displaystyle k\in \mathbb {N} } . For the place at infinity, this corresponds to the function field k ( ( 1 / t ) ) {\displaystyle k((1/t))} which are power series of the form ∑ n = − k ∞ a n ( 1 / t ) n {\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}} === Units === The integers have only two units, 1 and −1. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as ±i. The Eisenstein integers Z[exp(2πi / 3)] have six units. The integers in real quadratic number fields have infinitely many units. For example, in Z[√3], every power of 2 + √3 is a unit, and all these powers are distinct. In general, the group of units of O, denoted O×, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the roots of unity that lie in O. This group is cyclic. The free part is described by Dirichlet's unit theorem. This theorem says that rank of the free part is r1 + r2 − 1. Thus, for example, the only fields for which the rank of the free part is zero are Q and the imaginary quadratic fields. A more precise statement giving the structure of O× ⊗Z Q as a Galois module for the Galois group of K/Q is also possible. The free part of the unit group can be studied using the infinite places of K. Consider the function { L : K × → R r 1 + r 2 L ( x ) = ( log ⁡ \| x \| v ) v {\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log \|x\|_{v})_{v}\end{cases}}} where v varies over the infinite places of K and \|·\|v is the absolute value associated with v. The function L is a homomorphism from K× to a real vector space. It can be shown that the image of O× is a lattice that spans the hyperplane defined by x 1 + ⋯ + x r 1 + r 2 = 0. {\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.} The covolume of this lattice is the regulator of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the idele class group, that describes both the quotient by this lattice and the ideal class group. === Zeta function === The Dedekind zeta function of a number field, analogous to the Riemann zeta function, is an analytic object which describes the behavior of prime ideals in K. When K is an abelian extension of Q, Dedekind zeta functions are products of Dirichlet L-functions, with there being one factor for each Dirichlet character. The trivial character corresponds to the Riemann zeta function. When K is a Galois extension, the Dedekind zeta function is the Artin L-function of the regular representation of the Galois group of K, and it has a factorization in terms of irreducible Artin representations of the Galois group. The zeta function is related to the other invariants described above by the class number formula. === Local fields === Completing a number field K at a place w gives a complete field. If the valuation is Archimedean, one obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension K w / Q p : {\displaystyle K_{w}/\mathbf {Q} _{p}:} a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the Kronecker–Weber theorem can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry. == Major results == === Finiteness of the class group === One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer page 78. The order of the class group is called the class number, and is often denoted by the letter h. === Dirichlet's unit theorem === Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O. === Reciprocity laws === In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states ( p q ) ( q p ) = ( − 1 ) p − 1 2 q − 1 2 . {\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.} A reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin's reformulated reciprocity law states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see. === Class number formula === The class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. == Related areas == Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over Z instead of number rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds. == See also == Class field theory Kummer theory Locally compact field Tamagawa number == Notes == Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001 == Further reading == === Introductory texts === Stein, William (2012), Algebraic Number Theory, A Computational Approach (PDF) Ireland, Kenneth; Rosen, Michael (2013), A classical introduction to modern number theory, vol. 84, Springer, doi:10.1007/978-1-4757-2103-4, ISBN 978-1-4757-2103-4 Stewart, Ian; Tall, David (2015), Algebraic Number Theory and Fermat's Last Theorem, CRC Press, ISBN 978-1-4987-3840-8 === Intermediate texts === Marcus, Daniel A. (2018), Number Fields (2nd ed.), Springer, ISBN 978-3-319-90233-3 === Graduate level texts === Cassels, J. W. S.; Fröhlich, Albrecht, eds. (2010) [1967], Algebraic number theory (2nd ed.), London: 9780950273426, MR 0215665 Fröhlich, Albrecht; Taylor, Martin J. (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 0-521-43834-9, MR 1215934 Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, vol. 110 (2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR 1282723 Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. == External links == Media related to Algebraic number theory at Wikimedia Commons "Algebraic number theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Wikipedia:Algebra Colloquium#0	Algebra Colloquium is a journal founded in 1994. It was initially published by Springer-Verlag Hong Kong Ltd. In 2005, from volume 12 onwards, publishing rights were taken over by World Scientific. The company now publishes the journal quarterly. The journal is jointly edited by the Chinese Academy of Sciences and Soochow University. The journal mainly covers the field of pure and applied algebra. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.429. == Abstracting and indexing == Science Citation Index Expanded Research Alert CompuMath Citation Index MathSciNet Mathematical Reviews Zentralblatt MATH AJ VINITI (Russian) Chinese Science Citation Index Chinese Math Abstract == References == == External links == AC Journal Website
Wikipedia:Algebra Project#0	The Algebra Project is a national U.S. mathematics literacy program aimed at helping low-income students and students of color achieve the mathematical skills in high school that are a prerequisite for a college preparatory mathematics sequence. Founded by Civil Rights activist and Math educator Bob Moses in the 1980s, the Algebra Project provides curricular materials, teacher training, and professional development support and community involvement activities for schools to improve mathematics education. By 2001, the Algebra Project had trained approximately 300 teachers and was reaching 10,000 students in 28 locations in 10 states. == History == The Algebra Project was founded in 1982 by Bob Moses in Cambridge, Massachusetts. Moses worked with his daughter's eighth-grade teacher, Mary Lou Mehrling, to provide extra tutoring for several students in her class in algebra. Moses, who had taught secondary school mathematics in New York City and Tanzania, wanted to ensure that those students had sufficient algebra skills to qualify for honors math and science courses in high school. Through his tutorage, students from the Open Program of the Martin Luther King School passed the citywide algebra examination and qualified for ninth grade honors geometry, the first students from the program to do so. The Algebra Project grew out of attempts to recreate this on a wider community level, to provide similar students with a higher level of mathematical literacy. The Algebra Project now focuses on the southern states of the United States, where the Southern Initiative of the Algebra Project is directed by Dave Dennis. == Young People's Project == Founded in 1996, the Young People's Project (YPP) is a spin-off of the Algebra Project, which recruits and trains high school and college age "Math Literacy Workers" to tutor younger students in mathematics, and is directed by Omowale Moses. YPP has established sites in Jackson, Mississippi, Chicago, and the Greater Boston area of Massachusetts, and is developing programs in Miami, Petersburg, Virginia, Los Angeles, Ann Arbor, and Mansfield, Ohio. Each site employs between 30 and 100 high school and college age students part-time, and serves up to 1,000 elementary and middle-school students through on and off site programs. In 2005, the Algebra Project initiated Quality Education as a Civil Right (QECR), a national organizing effort to establish a federal constitutional guarantee of quality public education for all. Throughout 2005, YPP worked with students from Baltimore, New Orleans, Los Angeles, Oakland, Miami, Jackson, Chicago and Virginia to raise awareness about QECR. The Algebra Project and YPP students from Jackson and New Orleans hosted conferences, organized a Spring Break Community Education Tour to Miami and participated in QECR planning meetings at Howard University, the University of Michigan, and Jackson State University. == References == == External links == Algebra Project website Website of The Young People's Project Finding Our Folk Tour, a YPP Initiative NPR Codeswitch segment - To '60s Civil Rights Hero, Math Is Kids' Formula For Success
Wikipedia:Algebra Universalis#0	Algebra Universalis is an international scientific journal focused on universal algebra and lattice theory. The journal, founded in 1971 by George Grätzer, is currently published by Springer-Verlag. Honorary editors in chief of the journal included Alfred Tarski and Bjarni Jónsson. == External links == Algebra Universalis on Springer.com Algebra Universalis homepage, including instructions to authors Official website
Wikipedia:Algebra bundle#0	In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every algebra bundle is a vector bundle. Examples include the tensor-algebra bundle, exterior bundle, and symmetric bundle associated to a given vector bundle, as well as the Clifford bundle associated to any Riemannian vector bundle. == See also == Lie algebra bundle == References == Greub, Werner; Halperin, Stephen; Vanstone, Ray (1973), Connections, curvature, and cohomology. Vol. II: Lie groups, principal bundles, and characteristic classes, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, MR 0336651. Chidambara, C.; Kiranagi, B. S. (1994), "On cohomology of associative algebra bundles", Journal of the Ramanujan Mathematical Society, 9 (1): 1–12, MR 1279097. Kiranagi, B. S.; Rajendra, R. (2008), "Revisiting Hochschild cohomology for algebra bundles", Journal of Algebra and Its Applications, 7 (6): 685–715, doi:10.1142/S0219498808003041, MR 2483326. Kiranagi, B.S.; Ranjitha, Kumar; Prema, G. (2014), "On completely semisimple Lie algebra bundles", Journal of Algebra and Its Applications, 14 (2): 1–11, doi:10.1142/S0219498815500097.
Wikipedia:Algebra i Logika#0	Algebra i Logika (English: Algebra and Logic) is a peer-reviewed Russian mathematical journal founded in 1962 by Anatoly Ivanovich Malcev, published by the Siberian Fund for Algebra and Logic at Novosibirsk State University. An English translation of the journal is published by Springer-Verlag as Algebra and Logic since 1968. It published papers presented at the meetings of the "Algebra and Logic" seminar at the Novosibirsk State University. The journal is edited by academician Yury Yershov. The journal is reviewed cover-to-cover in Mathematical Reviews and Zentralblatt MATH. == Abstracting and Indexing == Algebra i Logika is indexed and abstracted in the following databases: According to the Journal Citation Reports, the journal had a 2020 impact factor of 0.753. == References == == External links == Algebra i Logika website Algebra and Logic website
Wikipedia:Algebra tile#0	Algebra tiles, also known as Algetiles, or Variable Blocks, are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students. They have also been used to prepare prison inmates for their General Educational Development (GED) tests. Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation. The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their Curriculum and Evaluation Standards for Mathematics. According to the NCTM 1989 standards "[r]elating models to one another builds a better understanding of each". == Examples == === Solving linear equations using addition === The linear equation x − 8 = 6 {\displaystyle x-8=6} can be modeled with one positive x {\displaystyle x} tile and eight negative unit tiles on the left side of a piece of paper and six positive unit tiles on the right side. To maintain equality of the sides, each action must be performed on both sides. For example, eight positive unit tiles can be added to both sides. Zero pairs of unit tiles are removed from the left side, leaving one positive x {\displaystyle x} tile. The right side has 14 positive unit tiles, so x = 14 {\displaystyle x=14} . === Solving linear equations using subtraction === The equation x + 7 = 10 {\displaystyle x+7=10} can be modeled with one positive x {\displaystyle x} tile and seven positive unit tiles on the left side and 10 positive unit tiles on the right side. Rather than adding the same number of tiles to both sides, the same number of tiles can be subtracted from both sides. For example, seven positive unit tiles can be removed from both sides. This leaves one positive x {\displaystyle x} tile on the left side and three positive unit tiles on the right side, so x = 3 {\displaystyle x=3} . === Multiplying polynomials === When using algebra tiles to multiply a monomial by a monomial, the student must first set up a rectangle where the length of the rectangle is the one monomial and then the width of the rectangle is the other monomial, similar to when one multiplies integers using algebra tiles. Once the sides of the rectangle are represented by the algebra tiles, one would then try to figure out which algebra tiles would fill in the rectangle. For instance, if one had x×x, the only algebra tile that would complete the rectangle would be x2, which is the answer. Multiplication of binomials is similar to multiplication of monomials when using the algebra tiles . Multiplication of binomials can also be thought of as creating a rectangle where the factors are the length and width. As with the monomials, one would set up the sides of the rectangle to be the factors and then fill in the rectangle with the algebra tiles. This method of using algebra tiles to multiply polynomials is known as the area model and it can also be applied to multiplying monomials and binomials with each other. An example of multiplying binomials is (2x+1)×(x+2) and the first step the student would take is set up two positive x tiles and one positive unit tile to represent the length of a rectangle and then one would take one positive x tile and two positive unit tiles to represent the width. These two lines of tiles would create a space that looks like a rectangle which can be filled in with certain tiles. In the case of this example the rectangle would be composed of two positive x2 tiles, five positive x tiles, and two positive unit tiles. So the solution is 2x2+5x+2. === Factoring === In order to factor using algebra tiles, one has to start out with a set of tiles that the student combines into a rectangle, this may require the use of adding zero pairs in order to make the rectangular shape. An example would be where one is given one positive x2 tile, three positive x tiles, and two positive unit tiles. The student forms the rectangle by having the x2 tile in the upper right corner, then one has two x tiles on the right side of the x2 tile, one x tile underneath the x2 tile, and two unit tiles are in the bottom right corner. By placing the algebra tiles to the sides of this rectangle we can determine that we need one positive x tile and one positive unit tile for the length and then one positive x tile and two positive unit tiles for the width. This means that the two factors are x + 1 {\displaystyle x+1} and x + 2 {\displaystyle x+2} . In a sense this is the reverse of the procedure for multiplying polynomials. == References == == Sources == Kitt, Nancy A. and Annette Ricks Leitze. "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts." MATHEMATICS TEACHER 2000. 462-520. Stein, Mary Kay et al., Implementing Standards-Based Mathematics Instruction. New York: Teachers College Press, 2000. Larson, Ronald E., Algebra 1. Illinois: McDougal Littell,1998. == External links == The National Library of Virtual Manipulatives
Wikipedia:Algebraic closure (convex analysis)#0	Algebraic closure of a subset A {\displaystyle A} of a vector space X {\displaystyle X} is the set of all points that are linearly accessible from A {\displaystyle A} . It is denoted by acl ⁡ A {\displaystyle \operatorname {acl} A} or acl X ⁡ A {\displaystyle \operatorname {acl} _{X}A} . A point x ∈ X {\displaystyle x\in X} is said to be linearly accessible from a subset A ⊆ X {\displaystyle A\subseteq X} if there exists some a ∈ A {\displaystyle a\in A} such that the line segment [ a , x ) := a + [ 0 , 1 ) ( x − a ) {\displaystyle [a,x):=a+[0,1)(x-a)} is contained in A {\displaystyle A} . Necessarily, A ⊆ acl ⁡ A ⊆ acl ⁡ acl ⁡ A ⊆ A ¯ {\displaystyle A\subseteq \operatorname {acl} A\subseteq \operatorname {acl} \operatorname {acl} A\subseteq {\overline {A}}} (the last inclusion holds when X is equipped by any vector topology, Hausdorff or not). The set A is algebraically closed if A = acl ⁡ A {\displaystyle A=\operatorname {acl} A} . The set acl ⁡ A ∖ aint ⁡ A {\displaystyle \operatorname {acl} A\setminus \operatorname {aint} A} is the algebraic boundary of A in X. == Examples == The set Q {\displaystyle \mathbb {Q} } of rational numbers is algebraically closed but Q c {\displaystyle \mathbb {Q} ^{c}} is not algebraically open If A = { ( x , y ) ∈ R 2 : 0 < y < x 2 } ⊆ R 2 {\displaystyle A=\{(x,y)\in \mathbb {R} ^{2}:0<y<x^{2}\}\subseteq \mathbb {R} ^{2}} then 0 ∈ ( acl ⁡ acl ⁡ A ) ∖ acl ⁡ A {\displaystyle 0\in (\operatorname {acl} \operatorname {acl} A)\setminus \operatorname {acl} A} . In particular, the algebraic closure need not be algebraically closed. Here, A ¯ = acl ⁡ acl ⁡ A = { ( x , y ) ∈ R 2 : 0 ≤ y ≤ x 2 } = ( acl ⁡ A ) ∪ { 0 } {\displaystyle {\overline {A}}=\operatorname {acl} \operatorname {acl} A=\{(x,y)\in \mathbb {R} ^{2}:0\leq y\leq x^{2}\}=(\operatorname {acl} A)\cup \{0\}} . However, acl ⁡ A = A ¯ {\displaystyle \operatorname {acl} A={\overline {A}}} for every finite-dimensional convex set A. Moreover, a convex set is algebraically closed if and only if its complement is algebraically open. == See also == Algebraic interior == References == == Bibliography == Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.

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