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https://mathoverflow.net/questions/8282 | 8 | For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\mathbb{CP}^2$ (or indeed) $\mathbb{CP}^n$) have a description in terms of line bundles. What happens in the Grassmannian sett... | https://mathoverflow.net/users/1648 | Holomorphic and antiholomorphic forms of projective space | Greg's otherwise excellent answer gives the impression that computing Chern classes on projective space requires a computer algebra system. I'm writing to repell this impression. The cohomology ring of $\mathbb{P}^{n-1}$ is $\mathbb{Z}[h]/h^n$ where $h$ is Poincare dual to the class of a hyperplane. We have the short e... | 7 | https://mathoverflow.net/users/297 | 8472 | 5,804 |
https://mathoverflow.net/questions/8468 | 10 | What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$?
I know about loop-erased random walk and uniform spanning trees.
What about conjectures in this direction? (Such as the double-dimer-cover cycles, which I read are conjectured to be $\text{SLE}(4)$)
I... | https://mathoverflow.net/users/2384 | Models with SLE scaling limit | There are several other models proved to converge to SLE: critical percolation on the triangular lattice, Gaussian Free Field, Harmonic Explorer, and recently also the critical Ising model. You can check the paper Kevin linked or [Schramm's slides](http://dbwilson.com/schramm/memorial/ICM.pdf) from ICM2006 for some hig... | 5 | https://mathoverflow.net/users/1061 | 8476 | 5,807 |
https://mathoverflow.net/questions/8260 | 46 | This question is related to (maybe even the same in intent as) [Intro to automatic theorem proving / logical foundations?](https://mathoverflow.net/questions/1017/intro-to-automatic-theorem-proving-logical-foundations), but none of the answers seem to address what I'm looking for.
There are a lot of resources availab... | https://mathoverflow.net/users/2383 | Proof assistants for mathematics | I am interested in the same kind of stuff. [This article](https://doi.org/10.1007/978-3-642-02614-0_34) tells about work done to formalize group representation theory in Coq.
In particular, they formalize the proof of Maschke's theorem (that $F[G]$ is semisimple when $G$ is a finite group).
Some links to math courses... | 10 | https://mathoverflow.net/users/400 | 8480 | 5,810 |
https://mathoverflow.net/questions/8483 | 8 | Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for rigid-analytic spaces that are the analog of commonly known theorems about complex analytic spaces?
| https://mathoverflow.net/users/2493 | Weierstrass points on rigid-analytic surfaces | Quick note: I am going to assume you want to talk about complete curves. One can, of course, have a curve with punctures in algebraic geometry, and I'm not sure how you'd want to define a Weierstrass point on it. In rigid geometry, you have even more freedom: you can have the analogue of a Riemann surface with holes of... | 9 | https://mathoverflow.net/users/297 | 8491 | 5,817 |
https://mathoverflow.net/questions/8477 | 6 | I recently launched a rocket with an altimeter that is accurate to roughly 10 ft. The recorded data is in time increments of 0.05 sec per sample and a graph of altitude vs. time looks pretty much like it should when zoomed out.
The problem is when I try to calculate other values such as velocity or acceleration from... | https://mathoverflow.net/users/2489 | Smoothing out Noisy Data | If you have data for the whole flight available to you then a good approach is Kalman smoothing. If you want estimates during the flight you want Kalman filtering. Seems like you're interested in the former. The difference is that Kalman smoothing uses data from the entire flight to estimate values at each point in tim... | 6 | https://mathoverflow.net/users/1233 | 8500 | 5,821 |
https://mathoverflow.net/questions/8473 | 7 | this is related to [this question](https://mathoverflow.net/questions/6627/convex-hull-in-cat0) but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be closed. I was wondering if anyone was aware of an explicit e... | https://mathoverflow.net/users/972 | Example of non-closed convex hull in a CAT(0) space | There are such examples already in Riemannian world!
In fact in any generic Riemannian manifold of dimension $\ge3$ convex hull of 3 points in general position is not closed.
BUT it is hard to make explicit and generic at the same time :)
If it is closed then there are a lot of geodesics lying in its boundary --- tha... | 10 | https://mathoverflow.net/users/1441 | 8505 | 5,825 |
https://mathoverflow.net/questions/8509 | 4 | Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots of examples. Thanks.
| https://mathoverflow.net/users/2231 | Martingales and Betting Strategies | I don't know about statistics *per se* but the best introduction to martingales *period* is Williams' *Probability with Martingales*:
[http://www.amazon.com/Probability-Martingales-Cambridge-Mathematical-Textbooks/dp/0521406056](http://rads.stackoverflow.com/amzn/click/0521406056)
| 14 | https://mathoverflow.net/users/1847 | 8510 | 5,827 |
https://mathoverflow.net/questions/8495 | 4 | What is the best way to find an actual divisor of an affine curve? I.E. if I am interested in finding a canonical divisor of a curve in two variables, is there a general way to go about it? Do I need to consider a projection on the x-axis?
I should clarify. I'm assuming the field is characteristic 0, and the curve i... | https://mathoverflow.net/users/2473 | Finding divisors on a curve | As others have pointed out, there are several different ways to explicitly write down a divisor. So it would be helpful to know what kind of answer you're looking for.
Anyhow, here's one answer. For a curve, the canonical divisor is the same as the sheaf of differential 1-forms. Let's assume that your curve $C$ in th... | 5 | https://mathoverflow.net/users/4 | 8511 | 5,828 |
https://mathoverflow.net/questions/8521 | 70 | As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive reason why a very simple proof is not possible?
| https://mathoverflow.net/users/2498 | Nice proof of the Jordan curve theorem? | There's a short proof (less than three pages) that uses Brouwer's fixed point theorem, available here:
[The Jordan Curve Theorem via the Brouwer Fixed Point Theorem](http://www.maths.ed.ac.uk/~aar/jordan/maehara.pdf)
The goal of the proof is to take Moise's "intuitive" proof and make it simpler/shorter. Not sure wh... | 35 | https://mathoverflow.net/users/1557 | 8522 | 5,832 |
https://mathoverflow.net/questions/8508 | 3 | Suppose we're given a strict and small 2-category $C$, and an object of $C$ called $A$. Can we produce an internal category structure on $A$ in some canonical way (maybe by some sort of argument similar to the construction of a group object [see my comment])?
If it's not possible in general, but it is possible in sp... | https://mathoverflow.net/users/1353 | Generation of object-internal structure in a strict 2-category | It's not entirely clear to me what you're asking, but I'll have a go at answering it anyway.
An internal category in a category with pullbacks consists of objects $A\_0$ and $A\_1$, maps $s,t\colon A\_1\to A\_0$, $i\colon A\_0\to A\_1$, and $c\colon A\_1\times\_{A\_0}A\_1\to A\_1$ satisfying the usual axioms. An equi... | 3 | https://mathoverflow.net/users/49 | 8526 | 5,835 |
https://mathoverflow.net/questions/8525 | 1 | We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured spaces.
In part 5 of Lurie's DAG, he describes this notion in terms of (infinity,1)-topoi. However, I don't see myself bei... | https://mathoverflow.net/users/1353 | Semiclassical explanation of "Structured" spaces | I don't quite understand what you want, but here's a shot in the dark: SGA 4 has a notion of ringed topos, and this is generalized in Lurie's work. Schemes and stacks have their "underlying topoi" with respect to whatever topology you put on them. I think you can define notions like smoothness and tangents in this sett... | 1 | https://mathoverflow.net/users/121 | 8527 | 5,836 |
https://mathoverflow.net/questions/8513 | 11 | This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If that's too hard, one could start off by assuming that the underlying space is a manifold. The example that got me thinking ... | https://mathoverflow.net/users/2497 | Characterization of Riemannian metrics | If $X$ is a metric space and $x$, $y\in X$, a *segment* from $x$ to $y$ is a subset $S\subseteq X$ such that $x$, $y\in S$ and $S$ is isometric to $[0,d(x,y)]$.
Let now $n\in\{1,2,3\}$ and let $X$ be a metric space which is locally compact, $n$-dimensional and such that *(i)* every two points are the endpoints of a ... | 19 | https://mathoverflow.net/users/1409 | 8535 | 5,841 |
https://mathoverflow.net/questions/8484 | 5 | What are the representations of $U(n)$ that induce (see [link text](https://mathoverflow.net/questions/5772/principal-bundles-representations-and-vector-bundles)) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known fact that $\mathbb{CP}^... | https://mathoverflow.net/users/1648 | Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space | The group U(n-1) has an abelian factor U(1) and a semisimple factor SU(n-1).
I'll answer first the second part of the question: Since line bundles are induced from one dimensional character representations, and the semismple component has no non-trivial character representations, thus the line bundles are induced fr... | 3 | https://mathoverflow.net/users/1059 | 8538 | 5,843 |
https://mathoverflow.net/questions/8494 | 9 | Maybe this is a silly question (or not even a question), but I was wondering whether the cotangent bundle of a submanifold is somehow **canonically** related to the cotangent bundle of the ambient space.
To be more precise:
Let $N$ be a manifold and $\iota:M \hookrightarrow N$ be an embedded (immersed) submanifold. ... | https://mathoverflow.net/users/675 | Cotangent bundle of a submanifold | It is possible to see the cotangent bundle of the submanifold as a kind of symplectic reduction of the cotangent bundle of the ambient manifold. I think it might be enough to explain the analogous fact from linear algebra.
Let V be a vector space and U a subspace. There is a natural symplectic form $\omega\_V$ on $V^... | 10 | https://mathoverflow.net/users/380 | 8555 | 5,853 |
https://mathoverflow.net/questions/8559 | 8 | Is it possible to find a coupling of two Wiener processes $W^0, W^x$ (i.e. two Wiener processes defined on a common probability space). One starting from $0$ and the other from $x$ such that
$W\_t^0 - W\_t^x \rightarrow\_t 0$ almost surely and in $L^1$.
Using some random-walks considerations I suspect that it is *... | https://mathoverflow.net/users/1302 | Coupling of Wiener processes | If there is no coupling s.t. the distance goes to 0 in $L^{1}$ (which I agree seems likely), you might want to look up an introduction to the Wasserstein-1 distance (which is exactly expected $L^{1}$ distance after an optimal coupling). This is the language that I've seen this type of problem most often discussed in. T... | 3 | https://mathoverflow.net/users/2282 | 8575 | 5,866 |
https://mathoverflow.net/questions/8579 | 38 | I remember learning some years ago of a theorem to the effect that if a polynomial $p(x\_1, ... x\_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of polynomials in the variables $x\_i$. Unfortunately, I'm not sure if I'm remembering correctly. (The context in which I saw this ... | https://mathoverflow.net/users/290 | Are all polynomial inequalities deducible from the trivial inequality? | One interpretation of the question is [Hilbert's seventeenth problem](https://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem), to characterize the polynomials on $\mathbb{R}^n$ that take non-negative values. The problem is motivated by the nice result, which is not very hard, that a non-negative polynomial in $\... | 42 | https://mathoverflow.net/users/1450 | 8582 | 5,871 |
https://mathoverflow.net/questions/5063 | 14 | Let $V$ be the irreducible representation of $GL\_n$ with highest $\lambda$, and $|\lambda|=n$. It is well known that the representation of $S\_n$ on the $(1,1,\ldots,1)$ weight space is the Specht module $S\_{\lambda}$. What is known about the representation of $S\_n$ on the rest of $V$?
I am also interested in the ... | https://mathoverflow.net/users/297 | Restriction from $GL_n$ to $S_n$ | I keep meaning to post this and forgetting: Richard Stanley sent me the following two references
*Enumerative Combinatorics, Volume II* Exercise 7.74: If $V\_{\lambda}$ is a representation of $GL\_n$, and $S\_{\mu}$ a representation of $S\_n$, then the multiplicity of $S\_{\mu}$ in $\mathrm{Restriction}^{GL\_n}\_{S\_... | 7 | https://mathoverflow.net/users/297 | 8589 | 5,876 |
https://mathoverflow.net/questions/8560 | 26 | Background: a field is *formally real* if -1 is not a sum of squares of elements in that field. An *ordering* on a field is a linear ordering which is (in exactly the sense that you would guess if you haven't seen this before) compatible with the field operations.
It is immediate to see that a field which can be orde... | https://mathoverflow.net/users/1149 | How much choice is needed to show that formally real fields can be ordered? | This is equivalent (in ZF) to the Boolean Prime Ideal Theorem (which is equivalent to the Ultrafilter Lemma).
Reference:
R. Berr, F. Delon, J. Schmid, Ordered fields and the ultrafilter theorem, Fund Math 159 (1999), 231-241. [online](http://matwbn.icm.edu.pl/ksiazki/fm/fm159/fm15933.pdf)
| 27 | https://mathoverflow.net/users/932 | 8591 | 5,878 |
https://mathoverflow.net/questions/8550 | 13 | Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory?
I suspect that this is true. The "operations" will be weighted sums, where the sum of the weights is at most $1$. The "free Banach space" on a set $X$ should be $\ell^1(X)$. (Note that the "und... | https://mathoverflow.net/users/45 | Is the category of Banach spaces with contractions an algebraic theory? | I think this is some kind of infinitary algebraic theory, but that it is *not* a monadic adjunction. That is, if you take the "closed unit ball functor" $B$ from ${\bf Ban}\\_1$ to ${\bf Set}$ and the "free Banach space functor" $L: {\bf Set} \to {\bf Ban}\_1$, then $L$ is left adjoint to $B$ but this adjunction is not... | 9 | https://mathoverflow.net/users/763 | 8593 | 5,880 |
https://mathoverflow.net/questions/8583 | -6 | I'm sorry if this isn't an appropriate question for MO. I've been reading here for a while, but I still haven't got a good grasp of what's a *good* question.
Given a field A and the polynomial ring A[x], we order the elements of A in any sequence and we define the isomorphism $f\colon A\to A[x]$ such that every el... | https://mathoverflow.net/users/1559 | Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]? | The question seems to involve a construction of a set-theoretic map, and the indexing (natural numbers?) suggests that A is assumed to have a countable underlying set. That map doesn't even yield a surjection of sets.
I would like to reinterpret the question in the following way: How much structure do we need to forg... | 5 | https://mathoverflow.net/users/121 | 8620 | 5,900 |
https://mathoverflow.net/questions/8608 | 19 | I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I've had is computing things like the join, product, coproduct, pullback, pushout, and so forth. I understand them as far as... | https://mathoverflow.net/users/1353 | Simplicial homotopy book suggestion for HTT computations | I'm not really sure what you are asking for. Colimits and limits are easy to compute in simplicial sets, because it's a presheaf category (as you say).
But if you want "geometrical" intuition about simplicial sets (including "pictures" of joins, etc.), you want to know about the geometric realization functor from simpl... | 14 | https://mathoverflow.net/users/437 | 8621 | 5,901 |
https://mathoverflow.net/questions/8632 | 12 | What is the cut rule? I don't mean the rule itself but an explanation of what it means and why are proof theorists always trying to eliminate it? Why is a cut-free system more special than one with cut?
| https://mathoverflow.net/users/nan | cut elimination | Suppose I have a proof of B starting from assumption A. And a proof of C starting from assumption B. Then the cut rule says I can deduce C from assumption A.
But I didn't need the cut rule. If I was able to deduce B from A I could simply "inline" the proof of B from A directly into the proof of C from B to get a proo... | 19 | https://mathoverflow.net/users/1233 | 8634 | 5,912 |
https://mathoverflow.net/questions/8599 | 35 | I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.
Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actuall... | https://mathoverflow.net/users/2515 | Tips on cohomology for number theory | Many number theorists, including me, learned Galois cohomology first via the proof of the Mordell-Weil theorem. The last chapter of Joe Silverman's book *The Arithmetic of Elliptic Curves* is a good source for this. It's very concrete and when you understand the proof you'll understand a lot about why number theorists ... | 17 | https://mathoverflow.net/users/431 | 8646 | 5,923 |
https://mathoverflow.net/questions/5303 | 20 | Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?
| https://mathoverflow.net/users/1172 | Basis of l^infinity | The question is about the complexity of the simplest possible Hamel basis of $\ell^\infty$, and this is a perfectly sensible thing to ask about even in a context where one wants to retain the Axiom of Choice. That is, we know by AC that there is a basis---how complex must it be?
Such a question finds a natural home i... | 25 | https://mathoverflow.net/users/1946 | 8647 | 5,924 |
https://mathoverflow.net/questions/8648 | 7 | For an affine variety, I know how to compute the set of singular points by simply looking at the points where the Jacobian matrix for the set of defining equations has too small a rank.
But what is the corresponding method for a variety that is a projective variety,and also a variety is a subset of a product of some... | https://mathoverflow.net/users/2623 | Easiest way to determine the singular locus of projective variety & resolution of singularities | The Jacobian condition for smoothness is valid also for projective varieties as well as affine varieties: you just take a homogeneous defining ideal and compute the rank of the Jacobian matrix at the point p, see e.g. p. 4 of
<http://www.ma.utexas.edu/users/gfarkas/teaching/alggeom/march4.pdf>
For a general variet... | 6 | https://mathoverflow.net/users/1149 | 8650 | 5,926 |
https://mathoverflow.net/questions/8597 | 13 | Answers can come in mathematical, physical, and philosophical flavors.
Edit: There seems to be a consensus that this question is not formulated well. I must respectfully disagree. My interest in the question is immaterial to the question itself. It is manifestly not a "what is" question. I see no reason to write more... | https://mathoverflow.net/users/2206 | What is the meaning of symplectic structure? | Here's how I understand it. Classical mechanics is done on a phase space *M*. If we are trying to describe a mechanical system with *n* particles, the phase space will be 6\*n\*-dimensional: 3\*n\* dimensions to describe the coordinates of particles, and 3\*n\* dimensions to describe the momenta. The most important pro... | 16 | https://mathoverflow.net/users/2467 | 8652 | 5,928 |
https://mathoverflow.net/questions/8667 | 7 | I want to ask a stupid question. Let $I$ be an infinite set and suppose $i$ belongs to $I$. I wonder whether following morphisms exist in general:
Hom($A$,colim $B\_i) \to$ lim Hom($A,B\_i$) and
Colim Hom($A,B\_i) \to$ Hom($A$,colim $B\_i$)
What I know is: if we replace lim by infinite product and colim by infin... | https://mathoverflow.net/users/1851 | On limits and Colimits | For any diagram $B\_i$ and an object $A$ in a category, there are natural maps of sets:
1. colim Hom($A,B\_i) \to$ Hom($A$, colim $B\_i$)
2. colim Hom($B\_i,A) \to$ Hom(lim $B\_i, A$)
These maps need not be isomorphisms, in general (neither even when the diagram is filtered, nor when it is finite). Nor are they iso... | 21 | https://mathoverflow.net/users/2106 | 8669 | 5,937 |
https://mathoverflow.net/questions/8671 | 15 | I am trying to get used to Hochschild cohomology of algebras by proving its properties. I am currently trying to show that the cup product is graded-commutative (because I heard this somewhere); however, my trouble is that I have no idea what the exact conditions are for this to hold.
As always in Hochschild cohomolo... | https://mathoverflow.net/users/2530 | Graded commutativity of cup in Hochschild cohomology | The cup product in Hochschild cohomology$H^\bullet(A,A)$ is graded commutative for all unitary algebras. If $M$ is an $A$-bimodule, then the cohomology $H^\bullet(A,M)$ with values in $M$ is a *symmetric* graded bimodule over $H^\bullet(A,A)$.
(If $M$ itself is also an algebra such that its multiplication map $M\otim... | 11 | https://mathoverflow.net/users/1409 | 8674 | 5,940 |
https://mathoverflow.net/questions/8663 | 10 | **Edit & Note:** I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category. Also, "model" in quotes means the English word model, whereas without quotes it has do do w... | https://mathoverflow.net/users/1353 | (infinity,1)-categories directly from model categories | You can't produce every ($\infty$,1)-category from a model category. The slogan is that every *presentable* ($\infty$,1)-category comes from a model category, and every adjoint pair between such comes from a Quillen pair of functors between model categories. The paper by Dugger on [Universal model categories](http://ar... | 25 | https://mathoverflow.net/users/437 | 8675 | 5,941 |
https://mathoverflow.net/questions/8665 | 29 | Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect that either it is known that there is such a diffeomorphism, or the problem is open. This is because if there was an embe... | https://mathoverflow.net/users/380 | Embeddings of $S^2$ in $\mathbb{CP}^2$ | The conjecture that every $S^2 \subseteq \mathbb{C}P^2$ is standard if it is homologous to flat is implied by the smooth Poincaré conjecture in 4 dimensions. It also implies a special of smooth Poincaré that is accepted as an open problem, the case of [Gluck surgery](http://atlas-conferences.com/c/a/b/o/02.htm) in $S^4... | 17 | https://mathoverflow.net/users/1450 | 8683 | 5,946 |
https://mathoverflow.net/questions/8684 | 18 | I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, binary products/coproducts, preimages,...
I know the (a?) definition of homotopy pullbacks/pushouts, but I am lacking tw... | https://mathoverflow.net/users/1797 | Homotopy pullbacks and homotopy pushouts | You can think of the pushout of two maps f : A → B, g : A → C in Set as computing the disjoint union of B and C with an identification f(a) = g(a) for each element a of A. We could imagine forming this as either the quotient by an equivalence relation, or by gluing in a segment joining f(a) to g(a) for each a, and taki... | 22 | https://mathoverflow.net/users/126667 | 8690 | 5,952 |
https://mathoverflow.net/questions/8656 | 2 | What is a "nice" way of choosing coset representatives for the symplectic group $Sp\_{2k}(\mathbb{C})$ in the general linear group $GL\_{2k}(\mathbb{C})$?
| https://mathoverflow.net/users/2623 | Nicest coset representatives of the symplectic group in the general linear group | I suppose that "nice" is very much application-dependent, but let me give it a try.
The first thing to notice is that, of course, you will not be able to find a global coset representative, since the principal bundle
$$\mathrm{Sp}(2k,\mathbb{C}) \to \mathrm{GL}(2k,\mathbb{C}) \to M = \mathrm{GL}(2k,\mathbb{C})/\mathr... | 3 | https://mathoverflow.net/users/394 | 8698 | 5,956 |
https://mathoverflow.net/questions/8692 | 15 | The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The *genus 0* GW invariants give the structure of quantum cohomology of $X$, which is then an example of a so-called Frobenius manifold. The mirror *genus 0* B-model theory on the mirror manifold (or ... | https://mathoverflow.net/users/83 | Higher genus closed string B-model | This is a great question I wish I understood the answer to better.
I know two vague answers, one based on derived algebraic geometry and one based on string theory.
The first answer, that Costello explained to me and I most likely misrepeat,
is the following. The B-model on a CY X as an extended TFT can be defined in ... | 15 | https://mathoverflow.net/users/582 | 8708 | 5,961 |
https://mathoverflow.net/questions/8697 | 13 | Is it true that any closed oriented $4$-dimensional manifold can be obtained as a result of the following construction:
Take $S^4$ with a finite collection of immersed closed 2-manifolds (with transversal intersections and self-intersections) and construct ramified cover of $S^4$ with a ramification of order at most... | https://mathoverflow.net/users/1441 | Ramified cover of 4-sphere | The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as $4$-fold branched covers of $S^4$. Topology 34 (1995), no. 3, 497--508. Any closed, orientable PL 4-manifold can be expressed as a 4-fold simple branched covering ... | 11 | https://mathoverflow.net/users/1822 | 8715 | 5,965 |
https://mathoverflow.net/questions/8716 | 11 | My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\mathbb{F}\_p^{n\times n}$ are similar to each other. Does this yield the existence of a matrix $X\in\mathrm{SL}\_n\left(\ma... | https://mathoverflow.net/users/2530 | Local-globalism for similar matrices? | Your question reminds me of a classical theorem of Latimer and MacDuffee (Annals of Math. 1933). To be sure, the theorem does not answer your question, but it seems relevant.
A nice contemporary treatment of this theorem (with references) can be found at
<http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/matrix... | 11 | https://mathoverflow.net/users/1149 | 8718 | 5,966 |
https://mathoverflow.net/questions/8723 | 6 | Is there a source in the literature for ribbon diagrams for the knot-table knots known to be ribbon knots?
For example, I'm interested in doing a computation which needs as input a ribbon diagram for the knot $8\_{20}$ (Rolfsen knot table notation). This knot is known to be ribbon, but I don't know a ribbon diagram f... | https://mathoverflow.net/users/1465 | Generating ribbon diagrams for knots known to be ribbon knots | I think Kawauchi's book has tables that include ribbon diagrams, but I don't have a copy with me. Look at
[Livingston and Cha](http://www.indiana.edu/~knotinfo/diagrams/8_20.png) . It is not hard to get a ribbon disk from this diagram: add a handle between the ears on the top and bottom right.
Generally, I check ... | 7 | https://mathoverflow.net/users/36108 | 8728 | 5,972 |
https://mathoverflow.net/questions/8741 | 15 | Here is a topic in the vein of [Describe a topic in one sentence](https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence "Describe a topic in one sentence") and [Fundamental examples](https://mathoverflow.net/questions/4994/fundamental-examples) : imagine that you are trying to explain and justify... | https://mathoverflow.net/users/2389 | Justifying a theory by a seemingly unrelated example | [In front of a blackboard, in an office at Real College]
Skeptic: And why should I care about holomorphic functions?
Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum\_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}$ ? Here $a$ is one of your cherished real numbers, but not an integer.
Skeptic: Well, hm...
Ho... | 38 | https://mathoverflow.net/users/450 | 8752 | 5,988 |
https://mathoverflow.net/questions/8756 | 29 | So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
EDIT: Of course $\overline{\mathbb{Q}} \neq \mathbb{C}$. I don't know what I was thinking.
| https://mathoverflow.net/users/96 | Examples of algebraic closures of finite index | Theorem (Artin-Schreier, 1927): Let K be an algebraically closed field and F a proper subfield of K with $[K:F] < \infty$. Then F is real-closed and $K = F(\sqrt{-1})$.
See e.g. Jacobson, Basic Algebra II, Theorem 11.14.
| 51 | https://mathoverflow.net/users/1149 | 8759 | 5,993 |
https://mathoverflow.net/questions/8707 | 15 | Take a closed $n$-manifold $M$ and fix a nice $n$-ball $B$ in $M$. How much information about $M$ does the set of knotted $(n-2)$-spheres of $B$ which are unknotted *in* $M$ give?
| https://mathoverflow.net/users/1409 | Knots that unknot in a manifold | Kevin Walker and a member of the forum (by e-mail) let me know about a flaw in the argument in my previous post. I'll erase it after putting up this post.
One of my claims was the if $K$ unknots in $M$ then it unknots in $B$, where $K$ is a knot in a ball $B \subset M$. If $K$ is the unknot in $M$ it bounds a disk $... | 6 | https://mathoverflow.net/users/1465 | 8786 | 6,014 |
https://mathoverflow.net/questions/8784 | 29 | Recall the following classical theorem of Cartan (!):
**Theorem (Lie III):** Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group.
Similarly, one can propose "Lie III" statements for Lie algebras over other fields, for super Lie algebras, for Lie algebroids, etc.
The p... | https://mathoverflow.net/users/78 | Why is Lie's Third Theorem difficult? | That gluing together of group chunks, constructed from the BCH formula is precisely more or less what Serre does to prove the theorem (in the first proof he gives in) his book on Lie groups and Lie algebras. [Serre, Jean-Pierre. Lie algebras and Lie groups. 1964 lectures given at Harvard University. Second edition. Lec... | 20 | https://mathoverflow.net/users/1409 | 8788 | 6,016 |
https://mathoverflow.net/questions/8789 | 89 | Let $M$ be a (real) manifold. Recall that an *analytic structure* on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is a structure, not a property.
Q1: Is it true that any topological mani... | https://mathoverflow.net/users/78 | Can every manifold be given an analytic structure? | (similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of Whitehead's. The lowest-dimensional examples of topological manifolds that don't admit triangulations are in dimension 4, th... | 87 | https://mathoverflow.net/users/1465 | 8794 | 6,019 |
https://mathoverflow.net/questions/8790 | 1 | I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a node at $(0,0)$ (and at some other points) (it's not a homework question, just a concept that I'm stuck on!).
$xy=x^6+y^6$.
Then the blow-up should be the closure of this set, taken over all $(x,y) \... | https://mathoverflow.net/users/2623 | finding the closure when blowing a variety at a singularity | Look at the affine pieces: over the open subset $u \neq 0$, you have a local coordinate $z = v/u$ and your equations can be written as $y = zx$ and $xy = x^6 + y^6$. Substituting $y$ in the second equation gives you $x^2z = x^6 + x^6 z^6$. Now this equation factors as $x^2 = 0$ and $z = x^4(1 + z^6)$; the locus where t... | 7 | https://mathoverflow.net/users/1797 | 8795 | 6,020 |
https://mathoverflow.net/questions/8793 | 4 | Recall that a *Lie group* is a group object in the category of *C*∞ manifolds.
If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure so that all the group operations are smooth? If so, how unique is this structure?
Is a continuous group-homomorphism be... | https://mathoverflow.net/users/78 | Is every group object in TopMan a Lie group? | As Greg Kuperberg indicated in the comments, this is Hilbert's 5th Problem. The answer is yes, a theorem of Gleason, Montgomery and Zippin from the 1950's.
| 5 | https://mathoverflow.net/users/1149 | 8796 | 6,021 |
https://mathoverflow.net/questions/8811 | 1 | Suppose you have a curve $C$ such that deg$K\_C =0$ and $\Gamma(C,\Omega\_C^1) \neq 0$. Does this automatically imply that $\vartheta\_C \equiv \Omega\_C^1$? My thought is yes, I've seen a proposition (Stanford AG course notes) that $\vartheta\_C \equiv \Omega\_C^1$ for a nonsingular plane cubic, but the proof is done ... | https://mathoverflow.net/users/2557 | Sheaf isomorphism. | On a complete nonsingular curve over an algebraically closed field, a line bundle of degree zero with a global section is necessarily the trivial bundle. This is lemma IV.1.2 in Hartshorne's *Algebraic Geometry*: if $\mathrm{deg} D = 0$ and $\mathcal{L} = O\_C(D)$ has a global section, then $D$ is linearly equivalent t... | 2 | https://mathoverflow.net/users/1797 | 8814 | 6,033 |
https://mathoverflow.net/questions/8753 | 7 | Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If so, why does it appear that nobody investigating it?
To contextualize this question, I refer to discovery of the equicon... | https://mathoverflow.net/users/nan | Alternative axiom to induction | A crucial difference between non-Euclidean geometries and "non-inductive" models of PA- is that any model of PA- contains a canonical copy of the true natural numbers, and in this copy of $N$, the induction schema is true. In other words, PA is part of the complete theory of a very canonical model of PA-, and as such i... | 15 | https://mathoverflow.net/users/93 | 8819 | 6,036 |
https://mathoverflow.net/questions/8812 | 17 | I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure.
But if the automorphism group of a variety has algebraic group structure, how do I know the automorphism group is an algebraic group.
For example, the automorphism group of an elliptic curve $A$ is an extension o... | https://mathoverflow.net/users/2348 | Why do automorphism groups of algebraic varieties have natural algebraic group structure? | It is not always true that the automorphism group of an algebraic variety has a natural algebraic group structure. For example, the automorphism group of $\mathbb{A}^2$ includes all the maps of the form $(x,y) \mapsto (x, y+f(x))$ where $f$ is any polynomial. I haven't thought through how to say this precisely in terms... | 17 | https://mathoverflow.net/users/297 | 8820 | 6,037 |
https://mathoverflow.net/questions/8816 | 2 | What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?
To clarify, an example.
Suppose that a bunch of people are playing a game with k (to start) weighted coins, such that heads appears with probability p < 1. When the players play a round, they flip all their... | https://mathoverflow.net/users/942 | Result of repeated applications of the binomial distribution? | Here's how I interpret your example: there are a bunch of coins (k initially), each being flipped every round until it comes up tails, at which point the coin is "out," And you want to know, after n rounds, the probability that exactly j coins are still active, for j = 0, ..., k. (The existence of multiple players seem... | 5 | https://mathoverflow.net/users/302 | 8826 | 6,041 |
https://mathoverflow.net/questions/8829 | 38 | Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that odd dimensional ones are boundaries of compact $(n+1)$-manifolds. My question is: are there any especially nice construct... | https://mathoverflow.net/users/1772 | What manifolds are bounded by RP^odd? | $\mathbb RP^3$ double-covers the lens space $L\_{4,1}$, so it's the boundary of the mapping cylinder of that covering map.
In general $\mathbb RP^n$ for $n$ odd double-covers such a lens space. So in general $\mathbb RP^n$ is the boundary of a pretty standard $I$-bundle over the appropriate lens space. To be specifi... | 41 | https://mathoverflow.net/users/1465 | 8830 | 6,044 |
https://mathoverflow.net/questions/8809 | 8 | Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that H=Stab(x)?
Before you jump out of your seat and say, "take X=G/H," let me point out that the question is basically equivalent† to "Is... | https://mathoverflow.net/users/1 | Is every subgroup of an algebraic group a stabilizer for some action? | In his book "Linear algebraic groups", 6.8, p98, Borel shows that the quotient of an affine algebraic group over a field by an algebraic subgroup exists as an algebraic variety, and he notes p.105 that Weil proved a similar result for arbitrary algebraic groups.
| 11 | https://mathoverflow.net/users/930 | 8831 | 6,045 |
https://mathoverflow.net/questions/8840 | 4 | You have $N$ boxes and $M$ balls. The $M$ balls are randomly distributed into the $N$ boxes. What is the expected number of empty boxes?
I came up with this formula:
$\sum\_{i=0}^{N}i\binom{N}{i}\left(\frac{N-i}{N}\right)^{M}$
This seems to yield the right answer. However, it requires calculating large numbers, s... | https://mathoverflow.net/users/1646 | Probability Question | Let $X\_i$ be a random variable with value 1 when box $i$ is empty and 0 otherwise. Now
$P(X\_i=1)=(1-\frac{1}{N})^{M}$. And the expected number of empty boxes is just $\mathbb{E}(\sum X\_i)=N\mathbb{E}(X\_1)\approx \frac{N}{e^M}$
EDIT: gave the answer in terms of M,N instead of the numerical values given originally.... | 5 | https://mathoverflow.net/users/2384 | 8843 | 6,053 |
https://mathoverflow.net/questions/4636 | 2 | There is a short section in the book [Locally Compact Groups](http://books.google.com/books?id=3_BPupMDRr8C&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false) by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending cha... | https://mathoverflow.net/users/1521 | Hausdorff Derived Series | The point of this section of Stroppel's book is to show, ultimately, that nothing new happens. Stroppel shows that each term in the Hausdorff derived series is nothing other than the closure of the same term in the usual derived series. A topological group is Hausdorff-solvable if and only if it is solvable, and the so... | 3 | https://mathoverflow.net/users/1450 | 8848 | 6,057 |
https://mathoverflow.net/questions/8817 | 4 | If L and M are two local systems on a space X, what can we say about the cohomology groups $H^i(X,L\otimes M)$ in terms of the cohomology of L and M? For example, can we determine their dimensions. You can assume X to be an affine connected smooth curve, if that helps.
For the same question for coherent locally free ... | https://mathoverflow.net/users/370 | cohomology groups of tensor product of sheaves | If $X$ can just be a topological space, there are examples in which $H^i(X;L)$ and $H^i(X;M)$ vanish entirely, but $H^i(X,;L \otimes M)$ does not. For example, in cohomology with real coefficients, maybe $X = \mathbb{R}P^3$ and $L = M$ is the local system corresponding to the tautological line bundle, or if you like th... | 14 | https://mathoverflow.net/users/1450 | 8855 | 6,061 |
https://mathoverflow.net/questions/8853 | 7 | In logic modules, theorems like Soundness and completeness of first order logic are proved. Later, Godel's incompleteness theorem is proved. May I ask what are assumed at the metalevel to prove such statements? It seems to me that whatever is assumed at the metalevel should not be more than whatever is being formulated... | https://mathoverflow.net/users/nan | What assumptions and methodology do metaproofs of logic theorems use and employ? | It depends on what you're trying to prove, and for what purpose you are proving these metatheorems.
So, the notion of "more" you're appealing to in asking about the metalevel is not completely well-defined. One common definition of the strength of a logic is "proof-theoretic strength", which basically amounts to the... | 8 | https://mathoverflow.net/users/1610 | 8862 | 6,067 |
https://mathoverflow.net/questions/8856 | 4 | For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) can be confusing.
Which domains, extension theorems and functions do you think are the most important for a learner to... | https://mathoverflow.net/users/nan | Most important domains, extension theorems, and functions in several complex variables | Here are a few points to guide you into the beautiful subject you had the good taste to choose.
1) Hartogs extension phenomenon :given two concentric balls in $ \mathbb C^n$, any holomorphic function $ B(0;M) \setminus B(0;m) \to \mathbb C$ extends to a holomorphic function $ B(0;M) \to \mathbb C$.This really launch... | 12 | https://mathoverflow.net/users/450 | 8864 | 6,068 |
https://mathoverflow.net/questions/8863 | 2 | I am looking for suitable algorithm to compute the eigenvalues and eigenvectors of a matrix. My matrix is sparse ( think of Finite Element Matrix), and it is very, very big ( think of hundreds of thousands or even million degrees of freedom).
The leading candidate for this task seems to be Lanczos algorithm.
The is... | https://mathoverflow.net/users/807 | The application of Lanczos Algorithm on sparse matrix | Lanczos is independent of the representation of your matrix. It does not store or operate on the entries of your matrix. The input to the algorithm is not the matrix $A$ itself, but a black-box subroutine for matrix-vector multiplication: you provide a method to compute $Av$ given vectors $v$. That's the only way it ne... | 10 | https://mathoverflow.net/users/302 | 8867 | 6,069 |
https://mathoverflow.net/questions/8870 | 7 | More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any algorithm that will eventually tell me whether the Hilbert Scheme is reduced or not there?
Just to make it harder, this... | https://mathoverflow.net/users/2363 | Is there a way to check if a relative Hilbert Scheme is reduced? | Showing nonreducedness of a Hilbert scheme is a hard question in general. The most direct algorithm would involve producing a Grobner basis for the defining ideal of the component in question, and then computing its radical and seeing if the two are equal. But with the exception of rather simple cases, this computation... | 7 | https://mathoverflow.net/users/4 | 8886 | 6,079 |
https://mathoverflow.net/questions/8887 | 10 | I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer:
Given an elliptic curve E defined over H, a number field, with complex multiplication by R, and P is a prime ideal in the maximal order of H and E has good reduction at P. Is it le... | https://mathoverflow.net/users/2024 | Legitimacy of reducing mod p a complex multiplication action of an elliptic curve? | I'm not sure if there's a trivial way to see this. One answer is to
use the fact that every rational map from a variety X / $\mathbb{Z}\_p$ to an
abelian scheme is actually defined on all of X (see for instance Milne's abelian
varieties notes). Here, since the generic fiber is open in X you can apply this
by viewing th... | 8 | https://mathoverflow.net/users/2 | 8891 | 6,081 |
https://mathoverflow.net/questions/8885 | 11 | Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$ has rank r. When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with everything defined over $\mathbb{Q}$, such that the group of $\mathbb{Q}$-rational sections of $S$ has rank at least r?
Edit:... | https://mathoverflow.net/users/1464 | Building elliptic curves into a family | If you require $C = P^1$ then it's probably not possible except for very small values of $r$. If you don't care about $C$, then here is something that might work.
Suppose $E$ is given by $y^2=x^3+ax+b$ and $P\_i=(x\_i,y\_i),i=1,\ldots,r$ is
a basis for the Mordell-Weil group. Let $C$ be the curve given by the system... | 10 | https://mathoverflow.net/users/2290 | 8894 | 6,083 |
https://mathoverflow.net/questions/8890 | 13 | Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?
The literature on such bundles over 3-manifolds is huge and my naive searches don't seem to turn up specific examples.
Roughly speaking, the Casson invariant counts flat bundles over 3-manifolds, s... | https://mathoverflow.net/users/380 | Flat SU(2) bundles over hyperbolic 3-manifolds | Many (compact orientable) hyperbolic 3-manifolds have non-trivial $SU(2)$ representations.
By Mostow rigidity, the representation of the fundamental group $\Gamma$ of a closed hyperbolic 3-manifold into $SL(2,\mathbb{C})$ (lifted from $PSL(2,\mathbb{C})$) may be conjugated so that it lies in $SL(2,K)$, for $K$ a num... | 18 | https://mathoverflow.net/users/1345 | 8897 | 6,086 |
https://mathoverflow.net/questions/8746 | 21 | Gale [famously showed](http://www.cs.cmu.edu/afs/cs/academic/class/15859-f01/www/notes/brouwer-hex.pdf) that the determinacy of n-player, n-dimensional Hex is equivalent to the Brouwer fixed point theorem in n dimensions.
We can (and Gale does) view this as saying that if you d-color the vertices of a certain graph s... | https://mathoverflow.net/users/382 | The density hex | For a closely related question when you do not insist that all non zero components of v-w has the same sign, then the answer is known: See the following paper: B. Bollobas, G. Kindler, I. Leader, and R. O'Donnell, Eliminating cycles in the discrete torus, LATIN 2006: Theoretical informatics, 202{210, Lecture Notes in C... | 6 | https://mathoverflow.net/users/1532 | 8900 | 6,089 |
https://mathoverflow.net/questions/8882 | 5 | Is it possible for SOME positive $c$, $c<1$ to find a pair of COMPACT hyperbolic manfiolds $M^3$ and $N^3$
with a positive degree map $$f: M^3 \to N^3,$$ such that $f$ is contacting with constant $c$? Are there may examples like this?
One can ask the same question of Riemann surfaces, and it seems to me that this sho... | https://mathoverflow.net/users/943 | Contracting maps of hyperbolic manifolds | In general, for any non-zero degree map from one closed negatively curved manifold to another, there is a canonical map (due to Besson-Courtois-Gallot) called the "natural map". However, it's only known to be pointwise volume decreasing, not necessarily contracting. They call this the "real Schwarz-Lemma". Applying the... | 5 | https://mathoverflow.net/users/1345 | 8904 | 6,091 |
https://mathoverflow.net/questions/8912 | 21 | I've been driven up a wall by the following question: let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what's the best upper bound one can prove on |p(1)|? (I only care about the asymptotic dependence on d and δ, not the constants.)
For ... | https://mathoverflow.net/users/2575 | Analogue of the Chebyshev polynomials over C? | I may be missing something obvious here. Let $f(z, z^{\*})$ be the polynomial in $z$ and $z^{\*}$ of degree $d$ which achieves $\exp(d \delta)$. Let $g(z)$ be the Laurent polynomial obtained from $f$ by replacing $z^{\*}$ by $z^{-1}$. On the unit circle, we have $f=g$.
Now, let $h$ be the polynomial $z^d g$. This is ... | 31 | https://mathoverflow.net/users/297 | 8915 | 6,097 |
https://mathoverflow.net/questions/8871 | 5 | Hi,
I see that the tetrad postulate:
$\nabla\_{\mu}e\_{\nu}^{I}=\partial\_{\mu}e\_{\nu}^{I}-\Gamma\_{\mu\nu}^{\rho}e\_{\rho}^{I}+\omega\_{\mu J}^{I}e\_{\nu}^{J}=0$
Can be merely derived from writing a tensor in two different basis (pure natural-coordinates $\{\partial\_\mu\}$ and mixed $\{\partial\_\mu\} + \{e\_a\}... | https://mathoverflow.net/users/2566 | Tetrad postulate: Implies or results from the metricity of the connection? | Having botched the first attempt at answering this question and not wanting to delete the evidence, let me try again here.
The "tetrad postulate" is independent from metricity and from the condition that the connection be torsion-free. It is simply the equivalence (via the vielbein) of two connections on two differen... | 0 | https://mathoverflow.net/users/394 | 8927 | 6,105 |
https://mathoverflow.net/questions/8918 | 37 | Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" because an algebraic space group which is not a scheme would be too awesome. Any group homomorphism from such a G to an alge... | https://mathoverflow.net/users/1 | Is an algebraic space group always a scheme? | I think the answer is yes. If S is a noetherian scheme and G is a relative algebraic group space over S, then there is a stratification of S such that over each stratum, G is a group scheme (see K. Behrend, Derived \ell-adic categories for Algebraic Stacks, 5.1.1). When S is Spec k there is no non-trivial stratificatio... | 15 | https://mathoverflow.net/users/370 | 8929 | 6,107 |
https://mathoverflow.net/questions/8926 | 7 | With regard to [my original question](https://mathoverflow.net/questions/7617/probability-question-closed):
>
> A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
>
>
>
I suppose that the responses that were elicited to my question wer... | https://mathoverflow.net/users/2582 | Probability vertices are adjacent in a polygon | It's not so hard to calculate the probability that no two points are adjacent:
We may as well assume that the first vertex is chosen for us. So let's ignore it and unroll the rest of the polygon into a line. Now imagine that I write down a sequence of $0$s and $1$s along this line to indicate whether the correspondin... | 5 | https://mathoverflow.net/users/2363 | 8934 | 6,109 |
https://mathoverflow.net/questions/8940 | 2 | f:X-->Y is flat and projective map between integral varieties over k, an algebraically closed field. Suppose every fiber at closed points of Y is still an integral variety. L is a line bundle on X, if f\_\*(L) is trivial line bundle on Y, and for every closed point y of Y, L\_y is also a trivial line bundle on fiber X\... | https://mathoverflow.net/users/2008 | Is line bundle determined by the parameter space and fiber? | Yes, you can. This follows from the see-saw principle for instance. You can also argue directly as follows. The spaces of global sections $H^{0}(Y,f\_\*L)$ and $H^{0}(X,L)$ are naturally isomorphic. Since $f\_\*L$ is a trivial line bundle we can choose a global nowhere vanishing section $e$ of $f\_\*L$. Let $s \in H^{0... | 6 | https://mathoverflow.net/users/439 | 8947 | 6,116 |
https://mathoverflow.net/questions/8907 | 4 | I am doing some research on the Spearman Rank Correlation Coefficient; all the references I can find refer essentially to a sample statistic. That is, given a *sample* of the jointly distributed $(x\_i,y\_i)$, one can compute the Spearman Coefficient between $x$ and $y$; I am wondering if there is a population equivale... | https://mathoverflow.net/users/2570 | Population Spearman Rank Correlation Coefficient | Let p(x,y) be the joint probability density function of the random variables X and Y. Let P\_x(x) and P\_y(y) the marginial cumulative distribution functions respectively. The key observation is that the normalized rank of a sample of x (i.e., its rank divided by the number of observations R(x\_i)/n) is just a sample o... | 1 | https://mathoverflow.net/users/1059 | 8955 | 6,122 |
https://mathoverflow.net/questions/8714 | 5 | It's well-known that, for lots of concrete categories (but [by no means all](https://mathoverflow.net/questions/1166/can-the-objects-of-every-concrete-category-themselves-be-realized-as-small-catego)), we can think of the objects as themselves being small categories, and morphisms are the functors between these categor... | https://mathoverflow.net/users/382 | What are natural transformations in 1-categories? | Here is a counterexample for your next-to-last question. Let S be a set with more than one element and consider the two full subcategories of Cat on, respectively, the single category which is the discrete category on S, and the single category which is the codiscrete category on S. In each case, when viewing Cat as a ... | 5 | https://mathoverflow.net/users/126667 | 8959 | 6,124 |
https://mathoverflow.net/questions/8957 | 144 | Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi\_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, perhaps like a more clever version of the proof that $\pi\_1(G)$ must be abelian?
| https://mathoverflow.net/users/2510 | Homotopy groups of Lie groups | I don't know of anything as bare hands as the proof that $\pi\_1(G)$ must be abelian, but here's a sketch proof I know (which can be found in Milnor's Morse Theory book. Plus, as an added bonus, one learns that $\pi\_3(G)$ has no torsion!):
First, (big theorem): Every (connected) Lie group deformation retracts onto i... | 144 | https://mathoverflow.net/users/1708 | 8961 | 6,126 |
https://mathoverflow.net/questions/8935 | 1 | When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection cohomology?
Someone was telling me something about this with shifting the IC or something, but I cannot remember the pre... | https://mathoverflow.net/users/2623 | intersection cohomology when the resolution is not semi-small | I'm not sure exactly what question you're asking. I think you may be looking for the following answer. By the decomposition theorem, the intersection cohomology of the variety is a direct summand of the cohomology of the resolution. I'm not sure there's anything more specific you can say than that in general.
| 1 | https://mathoverflow.net/users/916 | 8968 | 6,130 |
https://mathoverflow.net/questions/8972 | 38 | I have been thinking about which kind of wild non-measurable functions you can define. This led me to the question:
Is it possible to prove in ZFC, that if a (**Edit**: measurabel) set $A\subset \mathbb{R}$ has positive Lebesgue-measure, it has the same cardinality as $\mathbb{R}$? It is obvious if you assume CH, but... | https://mathoverflow.net/users/2097 | Do sets with positive Lebesgue measure have same cardinality as R? | I found the answer in the paper ["Measure and cardinality" by Briggs and Schaffter](http://www.jstor.org/pss/2320153). In short: not if I interpret positive measure to mean positive outer measure. A proof is given that every *measurable* subset with cardinality less than that of $\mathbb{R}$ has Lebesgue measure zero. ... | 42 | https://mathoverflow.net/users/1119 | 8975 | 6,134 |
https://mathoverflow.net/questions/8970 | 41 | I've heard that the problem of counting topologies is hard, but I couldn't really find anything about it on the rest of the internet. Has this problem been solved? If not, is there some feature that makes it pretty much intractable?
| https://mathoverflow.net/users/1353 | Number of valid topologies on a finite set of n elements | It's wiiiiiiide open to compute it exactly. As far as I know the "feature that makes it intractable" is that there's no real feature that makes it tractable. Very broadly speaking, if you want to count the ways that a generic type of structure can be put on an n-element set, there's no efficient way to do this -- you b... | 38 | https://mathoverflow.net/users/382 | 8980 | 6,138 |
https://mathoverflow.net/questions/8983 | 12 | Given a two-dimensional cubic Bézier spline defined by 4 control-points as described in [the Wikipedia entry](https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Cubic_B.C3.A9zier_curves), is there a way to solve analytically for the parameter along the curve (ranging from 0 to 1) which yields the point closest to an arbit... | https://mathoverflow.net/users/2201 | Closest point on Bézier spline | If you have a Bézier curve $(x(t),y(t))$, the closest point to the origin (say) is given by the minimum of $f(t) = x(t)^2 + y(t)^2$. By calculus, this minimum is either at the endpoints or when the derivative vanishes, $f'(t) = 0$. This latter condition is evidently a quintic polynomial. Now, there is no exact formula ... | 14 | https://mathoverflow.net/users/1450 | 8997 | 6,151 |
https://mathoverflow.net/questions/8999 | 15 | Given a graph $G$ we will call a function $f:V(G)\to \mathbb{R}$ discrete harmonic if for all $v\in V(G)$ , the value of $f(v)$ is equal to the average of the values of $f$ at all the neighbors of $v$. This is equivalent to saying the discrete Laplacian vanishes.
Discrete harmonic functions are sometimes used to appr... | https://mathoverflow.net/users/2384 | Discrete harmonic function on a planar graph | The answer is no.
I first describe the graph $G$. Let $N\_i$ be a sequence of positive integers; we will choose $N\_i$ later. Let $T$ be an infinite tree which has one root vertex, the root has $N\_1$ children; the children of that root have $N\_2$ children, those children have $N\_3$ children and so forth. Let $V\_... | 15 | https://mathoverflow.net/users/297 | 9003 | 6,154 |
https://mathoverflow.net/questions/8976 | 13 | The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itself). Then one could consider $\mathbf{Ord}+1$, $\mathbf{Ord}+\mathbf{Ord}$, $\mathbf{Ord} \cdot \mathbf{Ord}$ and so on. Does this extension ... | https://mathoverflow.net/users/158 | Ordinals that are not sets | Yes. This is both studied by set theorists and interesting. I personally find some of the related questions below extremely interesting, connected with some very deep questions about the nature of mathematical existence.
While the term "ordinal" is usually used only to refer to objects that are sets, one can neverthe... | 20 | https://mathoverflow.net/users/1946 | 9010 | 6,157 |
https://mathoverflow.net/questions/9007 | 16 | Let X be a topological space, let $\mathcal{U} = \{U\_i\}$ be a cover of X, and let $\mathcal{F}$ be a sheaf of abelian groups on X. If X is separated, each $U\_i$ is affine, and $\mathcal{F}$ is quasi-coherent, then Cech cohomology computes derived functor cohomology; in general one only gets a spectral sequence
$$
H^... | https://mathoverflow.net/users/2 | Cech to derived spectral sequence and sheafification | Yes, this is true in general.
It suffices to show the stalks vanish. Pick $x \in X$ and take an injective resolution $0 \to {\cal F} \to I^0 \to \cdots$. For any open $U$ containing $x$, we get a chain complex
$$0 \to I^0(U) \to I^1(U) \to \cdots$$
whose cohomology groups are $H^p(U,{\cal F}|\_U)$.
Taking direc... | 23 | https://mathoverflow.net/users/360 | 9012 | 6,158 |
https://mathoverflow.net/questions/8924 | 19 | Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder whether the surgeons' key problem has been solved. Is every simple homotopy equivalence between smooth, closed 3-manifolds... | https://mathoverflow.net/users/2356 | Diffeomorphism of 3-manifolds | Turaev [defined a simple-homotopy invariant](https://mathscinet.ams.org/mathscinet-getitem?mr=970081) which is a complete invariant of homeomorphism type (originally assuming geometrization).
Here is the Springer link if you have a subscription:
[Towards the topological classification of geometric 3-manifolds](https:... | 20 | https://mathoverflow.net/users/1345 | 9015 | 6,159 |
https://mathoverflow.net/questions/9022 | 9 | Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to figure out why. Is it obvious? A simple yes or no (with reference is possible) is all I need.
| https://mathoverflow.net/users/2612 | Quotient of a Hausdorff topological group by a closed subgroup | Edit: Below I expand my crude original answer "[Yes](http://en.wikipedia.org/wiki/Hausdorff_space)" as requested
by the community.
---
Yes. Let $G$ be the group and $H$ be the closed subgroup. The [kernel](http://en.wikipedia.org/wiki/Kernel_of_a_function) of the quotient map $G \to G/H$
is equal to $\Delta^{-1}... | 7 | https://mathoverflow.net/users/605 | 9023 | 6,163 |
https://mathoverflow.net/questions/9006 | 17 | EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it!
Background
----------
Let $E \to X$ be a holomorphic vector bundle over a complex manifold. A connection $A$ in $E$ is called *holomorphic* if in local holom... | https://mathoverflow.net/users/380 | Representations of surface groups via holomorphic connections | This question is addressed in a very recent paper of Bogomolov-Soloviev-Yotov (I don't think it is on the web yet). Among many interesting things they prove that the map from the moduli space of pairs $(C,\nabla)$ where $\nabla$ is a holomorphic connection on the trivial rank two bundle on some smooth curve $C$ is subm... | 13 | https://mathoverflow.net/users/439 | 9034 | 6,173 |
https://mathoverflow.net/questions/9031 | 8 | Is there a simple proof that 3-dimensional [graph manifolds](http://en.wikipedia.org/wiki/Graph_manifold) have residually finite fundamental groups?
By "simple" I mean the proof that does not use any hard 3d topology. I care because I wish to generalize this to higher-dimensional analogs of graph manifolds.
| https://mathoverflow.net/users/1573 | Residual finiteness for graph manifold groups | As far as I'm aware, every proof of this fact is essentially the same as [Hempel's original proof](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=AUCN&pg7=ALLF&pg8=ET&review_format=html&s4=hempel&s5=residual&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&y... | 9 | https://mathoverflow.net/users/1463 | 9036 | 6,175 |
https://mathoverflow.net/questions/8889 | 28 | I am not interested in the philosophical part of this question :-)
When I look at mathematics, I see that lots of different logics are used : classical, intuitionistic, linear, modal ones and weirder ones ...
For someone new to the field, it is not easy to really see what they have in common for justifying the use ... | https://mathoverflow.net/users/1173 | What is a logic? | I know nothing about this but I happened to come across it while reading about closure operators on wikipedia: [Universal Logic](http://en.wikipedia.org/wiki/Universal_logic).
| 12 | https://mathoverflow.net/users/1233 | 9042 | 6,180 |
https://mathoverflow.net/questions/9043 | 27 | This question is related to the question [Is an algebraic space group always a scheme?](https://mathoverflow.net/questions/8918/is-an-algebraic-space-group-always-a-scheme) which I've just seen which was posted by Anton. His question is whether an algebraic space which is a group object is necessarily a group scheme, a... | https://mathoverflow.net/users/184 | Why is this not an algebraic space? | If $X$ denotes the quotient sheaf $\mathbf{G}\_m / \mathbf{Z}$ then the inclusion
$\mathbf{G}\_m \times\_X \mathbf{G}\_m \rightarrow \mathbf{G}\_m \times \mathbf{G}\_m$
can be identified with the action map
$\mathbf{G}\_m \times \mathbf{Z} \rightarrow \mathbf{G}\_m \times \mathbf{G}\_m$.
Since this map is not ... | 21 | https://mathoverflow.net/users/32 | 9044 | 6,181 |
https://mathoverflow.net/questions/8948 | 2 | I've just finished my first course in differential geometry, so forgive me if this is maybe a silly or well-known question, but given any, say, diffeomorphism of $n$-manifolds $\phi:M\rightarrow N$, I was wondering whether the map $f:M\rightarrow GL\_n(\mathbb{R})$ defined by $f(p)=d\phi\_p$ has any important or nice p... | https://mathoverflow.net/users/1916 | Special map from a manifold to GL_n(R)? | The question can be interpreted to ask, given a diffeomorphism $\phi:M \to N$, what topological information in its derivative, interpreted as a bundle map? If the bundle map is used carefully, the answer is that there is a lot of information. There is a principle that, in dimension $n \ne 4$, there is enough informatio... | 9 | https://mathoverflow.net/users/1450 | 9057 | 6,191 |
https://mathoverflow.net/questions/9051 | 3 | I´m trying to solve a problem of cancellation of reflexive finitely generated modules over normal noetherian domains. When $R$ is regular domain with $\dim R \le 2$, for finitely generated modules, reflexive is equivalent to projective.
Now I´m studying the case $\dim R=2$ and $R$ normal. In this hypothesis, reflexi... | https://mathoverflow.net/users/2040 | About maximal Cohen-Macaulay modules | You probably want to look at this paper:
<http://www.springerlink.com/content/8r44x50448644568/>
on deformations of MCM modules and the references there.
| 4 | https://mathoverflow.net/users/2083 | 9062 | 6,195 |
https://mathoverflow.net/questions/9066 | 14 | Is that true that there is no rational curves contained in an Abelian variety? If it's true, is that because abelian varieties are not uniruled? How do I know whether an abelian variety is not uniruled?
| https://mathoverflow.net/users/2348 | Is there any rational curve on an Abelian variety? | There are no rational curves in an abelian variety, this is much stronger than not being uniruled. If there is a map $P^1 \to A$, $A$ abelian, the map would factor through the Albanese variety of $P^1$, by definition. However, for curves, the Albanese is the Jacobian (from general theory of the Jacobian) and the Jacobi... | 35 | https://mathoverflow.net/users/2290 | 9069 | 6,199 |
https://mathoverflow.net/questions/9065 | 10 | It's easy to see that in bipartite maximal triangle free graphs (n vertices), the maximum degree is at least $\lceil n/2 \rceil$. What about mtf graphs in general? Must there always be some vertex of high degree? Before I jump in with both feet, is there an obvious reason why there cannot be an absolute constant $c$ su... | https://mathoverflow.net/users/2588 | Maximum degree in maximal triangle free graphs | There is a sequence of [Kneser graphs](http://en.wikipedia.org/wiki/Kneser_graph), generalizing the [Petersen graph](http://en.wikipedia.org/wiki/Petersen_graph), that comprises a counterexample.
Let $k \ge 1$ be an integer and let $G$ be a graph whose vertices are subsets of size $k$ of $\{1,2,\ldots,3k-1\}$. Connec... | 9 | https://mathoverflow.net/users/1450 | 9070 | 6,200 |
https://mathoverflow.net/questions/9073 | 29 | Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not have a square root, but is the sum of at most 4 squares. (Lagrange Theorem). However, a real positive number has a square r... | https://mathoverflow.net/users/nan | When does 'positive' imply 'sum of squares'? | For many examples of this kind, see Olga Taussky, "Sums of squares", *Amer. Math. Monthly* 77 (1970) 805-830.
| 29 | https://mathoverflow.net/users/143 | 9075 | 6,204 |
https://mathoverflow.net/questions/4953 | 24 | Do we know any problem in NP which has a super-linear time complexity lower bound? Ideally, we would like to show that 3SAT has super-polynomial lower bounds, but I guess we're far away from that. I'd just like to know any examples of super-linear lower bounds.
I know that the time hierarchy theorem gives us problems... | https://mathoverflow.net/users/1042 | Super-linear time complexity lower bounds for any natural problem in NP? | Sorry I am so late to the discussion, but I just registered...
There are non-linear time lower bounds on multitape Turing machines for NP-complete problems. These lower bounds follow from the fact that the class of problems solvable in nondeterministic linear time is not equal to the class of those solvable in (deter... | 29 | https://mathoverflow.net/users/2618 | 9081 | 6,210 |
https://mathoverflow.net/questions/9013 | 4 | I don't know if this is an easy question for specialists in the field. Consider
the following interpolation problem : let $\varepsilon >0$, $X$ be a finite
set of real numbers and $g$ be a real-valued function
on $X$. The goal is to find a function $f$, defined on an interval
containing $X$, that coincides with $g$ on... | https://mathoverflow.net/users/2389 | Interpolation by a function whose second derivative is bounded | It isn't sufficient. Suppose that the elements of $X$ are
$$x\_1 < x\_2 < \cdots < x\_n,$$
and suppose for simplicity that $\epsilon = 1$. Then the values $f(x\_k)$ carry the same information as the integrals
$$\langle p, f'' \rangle = \int\_{x\_1}^{x\_n} p(x) f''(x) dx,$$
where $p(x)$ is a continuous, piecewise-linea... | 3 | https://mathoverflow.net/users/1450 | 9085 | 6,212 |
https://mathoverflow.net/questions/9055 | 3 | So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and its parameters in binary, the computer performs a bunch of boolean operations like $\wedge$, $\vee$, $\neg$ and gives you... | https://mathoverflow.net/users/nan | computation, algebra, logic |
>
> From this description it is not hard to see the connection of classical computation to discrete dynamical systems and classical logic, the operations provide the dynamics by flipping bits around in a controlled fashion and since we restrict the operations to a certain subset we get classical logic.
>
>
>
Car... | 7 | https://mathoverflow.net/users/1610 | 9096 | 6,222 |
https://mathoverflow.net/questions/9046 | 12 | Every commutative $C^\*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is the group algebra of some finite group. Does there exist a characterisation of the finitely generated commutative Hopf al... | https://mathoverflow.net/users/1095 | Hopf algebras arising as Group Algebras | Since the questioner starts off asking about $C^\*$ algebras, I am going to assume that he only cares about Hopf algebras over $\mathbb{C}$. **Every finitely generated, commutative $\mathbb{C}$-Hopf-algebra is the polynomial functions on an algebraic group $G$.** As Ben says, we can just take $G$ to be Spec of the Hopf... | 14 | https://mathoverflow.net/users/297 | 9104 | 6,226 |
https://mathoverflow.net/questions/9111 | 6 | Suppose I have a morphism of schemes for which I know the relative cotangent complex is trivial, and the map on reduced subschemes is an isomorphism. Is the map an isomorphism?
More generally, given a morphism of schemes with zero relative cotangent complex, which is of finite presentation on the reduced points. Is th... | https://mathoverflow.net/users/582 | Detecting etale maps on reduced points | Illusie, Complexe cotangent et deformations I, Prop. 3.1.1 (p. 203) is essentially the second thing you asked. Just a technical point: I don't think people would use the term "etale" unless the morphism is locally finitely presented or something like that (you seem to be wanting to assume that only at the level of redu... | 4 | https://mathoverflow.net/users/2628 | 9112 | 6,232 |
https://mathoverflow.net/questions/9100 | 21 | I'm currently working through Frenkel's beautiful paper:
<http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf>.
I'm looking for a good example of a projective curve to get my hands dirty, and go through the general constructions that Frenkel shows there and try to do them manually for this example of a curve. Are... | https://mathoverflow.net/users/2623 | A good example of a curve for geometric Langlands | Unfortunately I don't think geometric Langlands is very easy on any curve.
The only curve where the objects are readily accessible is $P^1$, but even there the general statement is kind of tricky (see Lafforgue's note [here](http://people.math.jussieu.fr/~vlafforg/geom.pdf)). I would look at Frenkel's writings on the G... | 21 | https://mathoverflow.net/users/582 | 9113 | 6,233 |
https://mathoverflow.net/questions/9000 | 22 | Wikipedia says that the [intermediate value theorem](https://en.wikipedia.org/wiki/Intermediate_value_theorem) “depends on (and is actually equivalent to) the completeness of the real numbers.” It then offers a simple counterexample to the analogous proposition on ℚ and a proof of the theorem in terms of the completene... | https://mathoverflow.net/users/2599 | Intermediate value theorem on computable reals | Let me assume that you are speaking about computable reals and functions in the sense of
[computable analysis](https://en.wikipedia.org/wiki/Computable_analysis), which is one of the most successful approaches to the topic. (One must be careful, since there are several incompatible notions of computability on the reals... | 31 | https://mathoverflow.net/users/1946 | 9116 | 6,235 |
https://mathoverflow.net/questions/9115 | 8 | Is there a notion of algebraic geometry for these objects? If we take the dual category of the category of cocommutative corings with counit, is there geometry in it in a sense dual to affine schemes? Can we look at the set of coideals of a coring, put a space structure on it and sheaves (maybe cosheaves) of sections?
... | https://mathoverflow.net/users/2300 | Algebraic geometry for cocommutative corings with counit. | Let's consider coalgebras over a field rather than corings. There is a theorem that every (coassociative) coalgebra over a field is the union of its finite-dimensional subcoalgebras. So the category of coalgebras over a field k is the category of ind-objects in the category of finite-dimensional coalgebras, while the l... | 15 | https://mathoverflow.net/users/2106 | 9121 | 6,240 |
https://mathoverflow.net/questions/9122 | 9 | A recursive presentation of a group is a one in which there is a finite number of generators and the set of relations is recursively enumerable. I found the following quote in Lyndon-Schupp, chapter II.1:
"This usage may seem a bit strange, but we shall see that if G has a presentation with the set of relations recur... | https://mathoverflow.net/users/2629 | Recursive presentations | The answer is a simple trick. Essentially no group theory is involved.
Suppose that we are given a group presentation with a set of generators, and relations R\_0, R\_1, etc. that have been given by a computably enumerable procedure. Let us view each relation as a word in the generators that is to become trivial in t... | 23 | https://mathoverflow.net/users/1946 | 9123 | 6,241 |
https://mathoverflow.net/questions/9089 | 10 | Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if the rational functions could have coefficients over Q rather than over R. Here is the relavant part of his speech
"At... | https://mathoverflow.net/users/nan | what was Hilbert's geometric construction in his 17th problem? | Actually the answer is in the sections 36 to 39 of Hilbert's "Foundations of geometry", which can be found on the web.
The constructions are construction with "straightedge" (ruler) and "transferrer of segments".
I quote a result from Hilbert's book :
Theorem 41. A problem in geometrical construction is, then, possi... | 10 | https://mathoverflow.net/users/2630 | 9124 | 6,242 |
https://mathoverflow.net/questions/9125 | 19 | The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; then the prime spectrum (which is equal to the maximal spectrum) is precisely the set of eigenvalues of $T$. The use of... | https://mathoverflow.net/users/290 | What is the origin of the term "spectrum" in mathematics? | Hilbert, in fact, got the term from Wilhelm Wirtinger (the first one to propose it according to, say <http://www.mathphysics.com/opthy/OpHistory.html>)
the paper of Wirtinger is "Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme" (1897).... | 29 | https://mathoverflow.net/users/2384 | 9126 | 6,243 |
https://mathoverflow.net/questions/9143 | 15 | In 'Cours d'arithmetique', Serre mentions in passing the following fact (communicated to him by Bombieri): Let P be the set of primes whose first (most significant) digit in decimal notation is 1. Then P possesses an analytic density, defined as
$\lim\_{s \to 1^+} \frac{\sum\_{p \in P} p^{-s}}{\log(\frac{1}{s-1})}$.
... | https://mathoverflow.net/users/25 | Analytic density of the set of primes starting with 1 | I think instead of posting my own explanation (which will only lose something in the translation) I'll instead refer you to two very interesting papers (thanks for posting this question, I haven't thought about this stuff in a couple years, and these papers were interesting reads to solve your problem.)
The first (am... | 13 | https://mathoverflow.net/users/2043 | 9145 | 6,255 |
https://mathoverflow.net/questions/8800 | 167 | K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature:
* Bott's original proof... | https://mathoverflow.net/users/1094 | Proofs of Bott periodicity | Here is my attempt to address Eric's actual question. Given a real $n$-dimensional vector bundle $E$ on a space $X$, there is an associated Thom space that can be understood as a twisted $n$-fold suspension $\Sigma^E X$. (If $E$ is trivial then it is a usual $n$-fold suspension $\Sigma^n X$.) In particular, if $E=L$ is... | 44 | https://mathoverflow.net/users/1450 | 9151 | 6,259 |
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