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In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them.
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https://en.wikipedia.org/wiki/Connection_one-form
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In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative. A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection.
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https://en.wikipedia.org/wiki/Connection_one-form
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The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.
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https://en.wikipedia.org/wiki/Connection_one-form
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In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into s-densities, whose coordinate representations become multiplied by the s-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T∗M (see pseudotensor).
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https://en.wikipedia.org/wiki/Density_bundle
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In mathematics, and specifically in functional analysis, the Lp sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right. The construction is motivated by the classical Lp spaces.
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https://en.wikipedia.org/wiki/Lp_sum
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In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6.
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https://en.wikipedia.org/wiki/Nonabelian_group
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It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.
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https://en.wikipedia.org/wiki/Nonabelian_group
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In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.
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https://en.wikipedia.org/wiki/Equivalent_measures
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In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
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https://en.wikipedia.org/wiki/Sum-of-divisors_function
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In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.
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https://en.wikipedia.org/wiki/Positive-definite_function_on_a_group
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In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.
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https://en.wikipedia.org/wiki/Poisson_kernel
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In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy excision theorem.
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https://en.wikipedia.org/wiki/Freudenthal_suspension_theorem
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In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Rooted graphs may also be known (depending on their application) as pointed graphs or flow graphs. In some of the applications of these graphs, there is an additional requirement that the whole graph be reachable from the root vertex.
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https://en.wikipedia.org/wiki/Accessible_pointed_graph
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In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together, contain all the real roots of the polynomial. Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root. Some algorithms compute all complex roots, but, as there are generally much fewer real roots than complex roots, most of their computation time is generally spent for computing non-real roots (in the average, a polynomial of degree n has n complex roots, and only log n real roots; see Geometrical properties of polynomial roots § Real roots).
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https://en.wikipedia.org/wiki/Real-root_isolation
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Moreover, it may be difficult to distinguish the real roots from the non-real roots with small imaginary part (see the example of Wilkinson's polynomial in next section). The first complete real-root isolation algorithm results from Sturm's theorem (1829). However, when real-root-isolation algorithms began to be implemented on computers it appeared that algorithms derived from Sturm's theorem are less efficient than those derived from Descartes' rule of signs (1637). Since the beginning of 20th century there is an active research activity for improving the algorithms derived from Descartes' rule of signs, getting very efficient implementations, and computing their computational complexity. The best implementations can routinely isolate real roots of polynomials of degree more than 1,000.
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https://en.wikipedia.org/wiki/Real-root_isolation
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In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments.
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https://en.wikipedia.org/wiki/Anti-commutative_property
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In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to z exists in the neighbourhood of each and every point in that set, where z is the complex conjugate. A definition of antiholomorphic function follows: " function f ( z ) = u + i v {\displaystyle f(z)=u+iv} of one or more complex variables z = ( z 1 , … , z n ) ∈ C n {\displaystyle z=\left(z_{1},\dots ,z_{n}\right)\in \mathbb {C} ^{n}} is the complex conjugate of a holomorphic function f ( z ) ¯ = u − i v {\displaystyle {\overline {f\left(z\right)}}=u-iv} ." One can show that if f(z) is a holomorphic function on an open set D, then f(z) is an antiholomorphic function on D, where D is the reflection against the x-axis of D, or in other words, D is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function.
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https://en.wikipedia.org/wiki/Antiholomorphic_function
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This implies that a function is antiholomorphic if and only if it can be expanded in a power series in z in a neighborhood of each point in its domain. Also, a function f(z) is antiholomorphic on an open set D if and only if the function f(z) is holomorphic on D. If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain. == References ==
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https://en.wikipedia.org/wiki/Antiholomorphic_function
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In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. A gauge symmetry of a Lagrangian L {\displaystyle L} is defined as a differential operator on some vector bundle E {\displaystyle E} taking its values in the linear space of (variational or exact) symmetries of L {\displaystyle L} . Therefore, a gauge symmetry of L {\displaystyle L} depends on sections of E {\displaystyle E} and their partial derivatives.
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https://en.wikipedia.org/wiki/Gauge_symmetry_(mathematics)
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For instance, this is the case of gauge symmetries in classical field theory. Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.Gauge symmetries possess the following two peculiarities. Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current J μ {\displaystyle J^{\mu }} takes a particular superpotential form J μ = W μ + d ν U ν μ {\displaystyle J^{\mu }=W^{\mu }+d_{\nu }U^{\nu \mu }} where the first term W μ {\displaystyle W^{\mu }} vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where U ν μ {\displaystyle U^{\nu \mu }} is called a superpotential. In accordance with Noether's second theorem, there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.Note that, in quantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.
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https://en.wikipedia.org/wiki/Gauge_symmetry_(mathematics)
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In mathematics, any integrable function s ( t ) {\displaystyle s(t)} can be made into a periodic function s P ( t ) {\displaystyle s_{P}(t)} with period P by summing the translations of the function s ( t ) {\displaystyle s(t)} by integer multiples of P. This is called periodic summation: s P ( t ) = ∑ n = − ∞ ∞ s ( t + n P ) {\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)} When s P ( t ) {\displaystyle s_{P}(t)} is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, S ( f ) ≜ F { s ( t ) } , {\displaystyle S(f)\triangleq {\mathcal {F}}\{s(t)\},} at intervals of 1 P {\displaystyle {\tfrac {1}{P}}} . That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s ( t ) {\displaystyle s(t)} at constant intervals (T) is equivalent to a periodic summation of S ( f ) , {\displaystyle S(f),} which is known as a discrete-time Fourier transform. The periodic summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.
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https://en.wikipedia.org/wiki/Periodic_summation
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In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces.
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https://en.wikipedia.org/wiki/Topological_dual_space
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When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for dual include polarer Raum , espace conjugué, adjoint space , and transponierter Raum and . The term dual is due to Bourbaki 1938.
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https://en.wikipedia.org/wiki/Topological_dual_space
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In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible.
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https://en.wikipedia.org/wiki/Chebyshev_approximation
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This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated. Modern mathematical libraries often reduce the domain into many tiny segments and use a low-degree polynomial for each segment.
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https://en.wikipedia.org/wiki/Chebyshev_approximation
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In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
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https://en.wikipedia.org/wiki/Arithmetic_combinatorics
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In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers.
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https://en.wikipedia.org/wiki/Arithmetic_Geometry
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In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the nth term of an arithmetico-geometric sequence is the product of the nth term of an arithmetic sequence and the nth term of a geometric one. Arithmetico-geometric sequences arise in various applications, such as the computation of expected values in probability theory.
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https://en.wikipedia.org/wiki/Arithmetico–geometric_series
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For instance, the sequence 0 1 , 1 2 , 2 4 , 3 8 , 4 16 , 5 32 , ⋯ {\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{16}}},\ {\dfrac {\color {blue}{5}}{\color {green}{32}}},\cdots } is an arithmetico-geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green). The summation of this infinite sequence is known as an arithmetico-geometric series, and its most basic form has been called Gabriel's staircase: ∑ k = 1 ∞ k r k = r ( 1 − r ) 2 , f o r 0 < r < 1 {\displaystyle \sum _{k=1}^{\infty }{\color {blue}k}{\color {green}r^{k}}={\frac {r}{(1-r)^{2}}},\quad \mathrm {for\ } 0
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https://en.wikipedia.org/wiki/Arithmetico–geometric_series
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In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data. Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric
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https://en.wikipedia.org/wiki/Assembly_map
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In mathematics, asymmetry can arise in various ways. Examples include asymmetric relations, asymmetry of shapes in geometry, asymmetric graphs et cetera.
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https://en.wikipedia.org/wiki/Asymmetry
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In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf and arrow sheaf are represented by algebras. Because any such presheaf will have an associated site, we can consider quasi-coherent sheaves on the site, giving a topos-theoretic notion of modules. Duallypg 2, comodules over a Hopf algebroid are the purely algebraic analogue of this construction, giving a purely algebraic description of quasi-coherent sheaves on a stack: this is one of the first motivations behind the theory.
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https://en.wikipedia.org/wiki/Comodule_over_a_Hopf_algebroid
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In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions f: M → R {\displaystyle f\colon M\to \mathbb {R} } on a smooth manifold M {\displaystyle M} , their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. The theory is named after Jean Cerf, who initiated it in the late 1960s.
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https://en.wikipedia.org/wiki/Cerf_theory
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In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.
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https://en.wikipedia.org/wiki/Auxiliary_function
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In mathematics, axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e. axioms) that relate a number of primitive terms — in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms.
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https://en.wikipedia.org/wiki/Axiomatic_proof
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In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup. The Doi–Naganuma lifting from 1967 was a precursor of the base change lifting. Base change lifting was introduced by Hiroshi Saito (1975, 1975b, 1979) for Hilbert modular forms of cyclic totally real fields of prime degree, by comparing the trace of twisted Hecke operators on Hilbert modular forms with the trace of Hecke operators on ordinary modular forms. Shintani (1979) gave a representation theoretic interpretation of Saito's results and used this to generalize them. Langlands (1980) extended the base change lifting to more general automorphic forms and showed how to use the base change lifting for GL2 to prove the Artin conjecture for tetrahedral and some octahedral 2-dimensional representations of the Galois group. Gelbart (1977), Gérardin (1979) and Gérardin & Labesse (1979) gave expositions of the base change lifting for GL2 and its applications to the Artin conjecture.
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https://en.wikipedia.org/wiki/Base_change_lifting
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In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base. The basic hypergeometric series 2 ϕ 1 ( q α , q β ; q γ ; q , x ) {\displaystyle {}_{2}\phi _{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)} was first considered by Eduard Heine (1846). It becomes the hypergeometric series F ( α , β ; γ ; x ) {\displaystyle F(\alpha ,\beta ;\gamma ;x)} in the limit when base q = 1 {\displaystyle q=1} .
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https://en.wikipedia.org/wiki/Basic_hypergeometric_series
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In mathematics, biangular coordinates are a coordinate system for the plane where C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are two fixed points, and the position of a point P not on the line C 1 C 2 ¯ {\displaystyle {\overline {C_{1}C_{2}}}} is determined by the angles ∠ P C 1 C 2 {\displaystyle \angle PC_{1}C_{2}} and ∠ P C 2 C 1 . {\displaystyle \angle PC_{2}C_{1}.} The sine rule can be used to convert from biangular coordinates to two-center bipolar coordinates.
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https://en.wikipedia.org/wiki/Biangular_coordinates
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In mathematics, bicubic interpolation is an extension of cubic spline interpolation (a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular grid. The interpolated surface (meaning the kernel shape, not the image) is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.
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https://en.wikipedia.org/wiki/Bicubic_interpolation
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In image processing, bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling, when speed is not an issue. In contrast to bilinear interpolation, which only takes 4 pixels (2×2) into account, bicubic interpolation considers 16 pixels (4×4). Images resampled with bicubic interpolation can have different interpolation artifacts, depending on the b and c values chosen.
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https://en.wikipedia.org/wiki/Bicubic_interpolation
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In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals. Bilinear interpolation is performed using linear interpolation first in one direction, and then again in another direction. Although each step is linear in the sampled values and in the position, the interpolation as a whole is not linear but rather quadratic in the sample location. Bilinear interpolation is one of the basic resampling techniques in computer vision and image processing, where it is also called bilinear filtering or bilinear texture mapping.
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https://en.wikipedia.org/wiki/Bilinear_filtering
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In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points.
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https://en.wikipedia.org/wiki/Binary_splitting
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In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.
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https://en.wikipedia.org/wiki/Biracks_and_biquandles
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In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.
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https://en.wikipedia.org/wiki/Birational_automorphism
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In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets , braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions.Historically, other notations, such as the vinculum, were similarly used for grouping. In present-day use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius, and in 1629 by Albert Girard.
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https://en.wikipedia.org/wiki/Bracket_(mathematics)
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In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space R n {\displaystyle \mathbb {R} ^{n}} as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces. Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.
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https://en.wikipedia.org/wiki/Calculus_on_Euclidean_space
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In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges. This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be interpreted as discrete versions of partial differential equations or continuum variational models. Such equations and models are important tools to mathematically model, analyze, and process discrete information in many different research fields, e.g., image processing, machine learning, and network analysis. In applications, finite weighted graphs represent a finite number of entities by the graph's vertices, any pairwise relationships between these entities by graph edges, and the significance of a relationship by an edge weight function.
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https://en.wikipedia.org/wiki/Calculus_on_finite_weighted_graphs
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Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or community detection in a social network (where the vertices represent users of the network, the edges represent links between users, and the weight function indicates the strength of interactions between users). The main advantage of finite weighted graphs is that by not being restricted to highly regular structures such as discrete regular grids, lattice graphs, or meshes, they can be applied to represent abstract data with irregular interrelationships. If a finite weighted graph is geometrically embedded in a Euclidean space, i.e., the graph vertices represent points of this space, then it can be interpreted as a discrete approximation of a related nonlocal operator in the continuum setting.
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https://en.wikipedia.org/wiki/Calculus_on_finite_weighted_graphs
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In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.
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https://en.wikipedia.org/wiki/Canonical_singularity
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In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide. Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s.
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https://en.wikipedia.org/wiki/Catastrophe_theory
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It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system.
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https://en.wikipedia.org/wiki/Catastrophe_theory
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However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures. In the late 1970s, applications of catastrophe theory to areas outside its scope began to be criticized, especially in biology and social sciences. Zahler and Sussmann, in a 1977 article in Nature, referred to such applications as being "characterised by incorrect reasoning, far-fetched assumptions, erroneous consequences, and exaggerated claims". As a result, catastrophe theory has become less popular in applications.
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https://en.wikipedia.org/wiki/Catastrophe_theory
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In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.
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https://en.wikipedia.org/wiki/Categorification
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The reverse of categorification is the process of decategorification. Decategorification is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the representation theory of Lie algebras, modules over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'.
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https://en.wikipedia.org/wiki/Categorification
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In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.
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https://en.wikipedia.org/wiki/Cellular_homology
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In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
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https://en.wikipedia.org/wiki/Derived_functors
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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.
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https://en.wikipedia.org/wiki/Mathematical_fallacies
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Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules. The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy.
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https://en.wikipedia.org/wiki/Mathematical_fallacies
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The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory).
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https://en.wikipedia.org/wiki/Mathematical_fallacies
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Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.
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https://en.wikipedia.org/wiki/Mathematical_fallacies
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In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems.
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https://en.wikipedia.org/wiki/Integral_manifold
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A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system). Given a collection of differential 1-forms α i , i = 1 , 2 , … , k {\displaystyle \textstyle \alpha _{i},i=1,2,\dots ,k} on an n {\displaystyle \textstyle n} -dimensional manifold M {\displaystyle M} , an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point p ∈ N {\displaystyle \textstyle p\in N} is annihilated by (the pullback of) each α i {\displaystyle \textstyle \alpha _{i}} .
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https://en.wikipedia.org/wiki/Integral_manifold
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A maximal integral manifold is an immersed (not necessarily embedded) submanifold i: N ⊂ M {\displaystyle i:N\subset M} such that the kernel of the restriction map on forms i ∗: Ω p 1 ( M ) → Ω p 1 ( N ) {\displaystyle i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)} is spanned by the α i {\displaystyle \textstyle \alpha _{i}} at every point p {\displaystyle p} of N {\displaystyle N} . If in addition the α i {\displaystyle \textstyle \alpha _{i}} are linearly independent, then N {\displaystyle N} is ( n − k {\displaystyle n-k} )-dimensional. A Pfaffian system is said to be completely integrable if M {\displaystyle M} admits a foliation by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.) An integrability condition is a condition on the α i {\displaystyle \alpha _{i}} to guarantee that there will be integral submanifolds of sufficiently high dimension.
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https://en.wikipedia.org/wiki/Integral_manifold
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In mathematics, chiral homology, introduced by Alexander Beilinson and Vladimir Drinfeld, is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine D X {\displaystyle {\mathcal {D}}_{X}} -scheme (i.e., the space of global solutions of a system of non-linear differential equations)." Jacob Lurie's topological chiral homology gives an analog for manifolds.
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https://en.wikipedia.org/wiki/Chiral_homology
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In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.
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https://en.wikipedia.org/wiki/Chromatic_tower
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In mathematics, circle-valued Morse theory studies the topology of a smooth manifold by analyzing the critical points of smooth maps from the manifold to the circle, in the framework of Morse homology. It is an important special case of Sergei Novikov's Morse theory of closed one-forms.Michael Hutchings and Yi-Jen Lee have connected it to Reidemeister torsion and Seiberg–Witten theory. == References ==
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https://en.wikipedia.org/wiki/Circle-valued_Morse_theory
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In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism θ L / F: C F / N L / F ( C L ) → Gal ( L / F ) , {\displaystyle \theta _{L/F}:C_{F}/{N_{L/F}(C_{L})}\to \operatorname {Gal} (L/F),} where N L / F {\displaystyle N_{L/F}} denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.
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https://en.wikipedia.org/wiki/Global_class_field_theory
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The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of C F . {\displaystyle C_{F}.} A standard method for developing global class field theory since the 1930s was to construct local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory.
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https://en.wikipedia.org/wiki/Global_class_field_theory
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This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic.
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https://en.wikipedia.org/wiki/Global_class_field_theory
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Inside class field theory one can distinguish special class field theory and general class field theory. Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of Q {\displaystyle \mathbb {Q} } , and the theory of complex multiplication to construct abelian extensions of CM-fields. There are three main generalizations of class field theory: higher class field theory, the Langlands program (or 'Langlands correspondences'), and anabelian geometry.
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https://en.wikipedia.org/wiki/Global_class_field_theory
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In mathematics, class field theory is the study of abelian extensions of local and global fields.
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https://en.wikipedia.org/wiki/Timeline_of_class_field_theory
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In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
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https://en.wikipedia.org/wiki/Classical_Wiener_space
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In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way.
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https://en.wikipedia.org/wiki/Sweedler_notation
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In finite dimensions, this duality goes in both directions (see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science.
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https://en.wikipedia.org/wiki/Sweedler_notation
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In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous. Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale.
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https://en.wikipedia.org/wiki/Coarse_function
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In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher. The boundary of an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries.
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https://en.wikipedia.org/wiki/Oriented_cobordant
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The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, ∂ W = M ⊔ N {\displaystyle \partial W=M\sqcup N} .
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https://en.wikipedia.org/wiki/Oriented_cobordant
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Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism.
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https://en.wikipedia.org/wiki/Oriented_cobordant
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Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.
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https://en.wikipedia.org/wiki/Oriented_cobordant
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In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name (Lemma 6),(Lemma 2.5),(Theorem 1), or by ad-hoc monikers such as vanishing lemma or inverse embedding.Cocompactness property allows to verify convergence of sequences, based on translational or scaling invariance in the problem, and is usually considered in the context of Sobolev spaces. The term cocompact embedding is inspired by the notion of cocompact topological space.
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https://en.wikipedia.org/wiki/Cocompact_embedding
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In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension.
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https://en.wikipedia.org/wiki/Dimension_counting
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In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point.
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https://en.wikipedia.org/wiki/Coherent_duality
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The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, Residues and Duality (1966) by Robin Hartshorne, became a reference.
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https://en.wikipedia.org/wiki/Coherent_duality
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One concrete spin-off was the Grothendieck residue. To go beyond proper morphisms, as for the versions of Poincaré duality that are not for closed manifolds, requires some version of the compact support concept. This was addressed in SGA2 in terms of local cohomology, and Grothendieck local duality; and subsequently. The Greenlees–May duality, first formulated in 1976 by Ralf Strebel and in 1978 by Eben Matlis, is part of the continuing consideration of this area.
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https://en.wikipedia.org/wiki/Coherent_duality
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In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
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https://en.wikipedia.org/wiki/Cohomology_with_compact_support
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In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.
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https://en.wikipedia.org/wiki/Combinatorial_group_theory
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In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence.
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https://en.wikipedia.org/wiki/Combinatorial_topology
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The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods.
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https://en.wikipedia.org/wiki/Combinatorial_topology
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Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).
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https://en.wikipedia.org/wiki/Combinatorial_topology
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In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.
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https://en.wikipedia.org/wiki/Compact_object_(category_theory)
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In mathematics, compactly generated can refer to: Compactly generated group, a topological group which is algebraically generated by one of its compact subsets Compactly generated space, a topological space whose topology is coherent with the family of all compact subspaces
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https://en.wikipedia.org/wiki/Compactly_generated
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In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.
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https://en.wikipedia.org/wiki/Comparison_theorem
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In mathematics, complementary series representations of a reductive real or p-adic Lie groups are certain irreducible unitary representations that are not tempered and do not appear in the decomposition of the regular representation into irreducible representations. They are rather mysterious: they do not turn up very often, and seem to exist by accident. They were sometimes overlooked, in fact, in some earlier claims to have classified the irreducible unitary representations of certain groups.
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https://en.wikipedia.org/wiki/Stein_complementary_series_representation
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Several conjectures in mathematics, such as the Selberg conjecture, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R). Elias M. Stein (1972) constructed some families of them for higher rank groups using analytic continuation, sometimes called the Stein complementary series.
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https://en.wikipedia.org/wiki/Stein_complementary_series_representation
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In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute. The generalized homology and cohomology complex cobordism theories were introduced by Michael Atiyah (1961) using the Thom spectrum.
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https://en.wikipedia.org/wiki/Complex_cobordism_ring
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In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a Cartesian product of the form C d {\displaystyle \mathbb {C} ^{d}} for some d {\displaystyle d} , and the complex dimension is the exponent d {\displaystyle d} in this product. Because C {\displaystyle \mathbb {C} } can in turn be modeled by R 2 {\displaystyle \mathbb {R} ^{2}} , a space with complex dimension d {\displaystyle d} will have real dimension 2 d {\displaystyle 2d} . That is, a smooth manifold of complex dimension d {\displaystyle d} has real dimension 2 d {\displaystyle 2d} ; and a complex algebraic variety of complex dimension d {\displaystyle d} , away from any singular point, will also be a smooth manifold of real dimension 2 d {\displaystyle 2d} .
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https://en.wikipedia.org/wiki/Real_dimension
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However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points. The real dimension is not greater than the dimension, and equals it if the variety is irreducible and has real points that are nonsingular. For example, the equation x 2 + y 2 + z 2 = 0 {\displaystyle x^{2}+y^{2}+z^{2}=0} defines a variety of (complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.The same considerations apply to codimension.
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https://en.wikipedia.org/wiki/Real_dimension
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For example a smooth complex hypersurface in complex projective space of dimension n will be a manifold of dimension 2(n − 1). A complex hyperplane does not separate a complex projective space into two components, because it has real codimension 2. == References ==
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https://en.wikipedia.org/wiki/Real_dimension
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In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas.
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https://en.wikipedia.org/wiki/Complex_geometry
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Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces sets the field apart from differential geometry, where the classification of possible smooth manifolds is a significantly harder problem.
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https://en.wikipedia.org/wiki/Complex_geometry
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