All Questions

5
votes
3answers
2k views

Is a torsion free abelian group finitely generated, if all of its localizations at primes p are finitely generated over Zp?

Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{...
3
votes
2answers
328 views

Definition modifications without choice

What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...
19
votes
6answers
2k views

Quantitative versions of ergodic theorem

Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...
11
votes
2answers
518 views

What do decategorification and “compactification on a circle” have to do with each other?

Some physicists have told me that if you think about an extended n-dimensional TQFT $F$, then the decategorification is given by $F'(X)=F(X\times S^1)$, which I believe they call "compactification on ...
8
votes
5answers
2k views

Pronunciation: Crapo

A similar question reminds me: When giving talks, I often want to refer to the work of Henry Crapo. I have asked several mathematicians, and none of them were sure how to pronounce his last name. Any ...
2
votes
1answer
27k views

Pronunciation: Dijkstra [closed]

I know how to pronounce Dijkstra's name correctly (hear it here: http://en.wikipedia.org/wiki/Edsger_W._Dijkstra). But I'd like to know how people usually say his name. I've heard it in many ...
13
votes
1answer
2k views

Solvable class field theory

Is/should there be a theory of finite solvable extensions over a given base field? Could it be based on/use class field theory? Assume the base field isn't a local field.
0
votes
2answers
240 views

What is the simplest non-recursive formulation for the following recursive function?

C(0) = 1 C(1) = 1 C(n+1) = Sigma(r, 0, n, C(r) x C(n-r)) Where Sigma() means: Sigma(index var, lower bound, upper bound (inclusive), function(r)) I'm not familiar ...
5
votes
5answers
2k views

Texts In Non-Commutative Harmonic Analysis

What texts/books are available for progressing into non-commutative harmonic analysis?
15
votes
6answers
4k views

Cohomology of fibrations over the circle

Are there any general results on the (integral) cohomology of manifolds that are fibrations over the circle? Any literature references much appreciated.
2
votes
3answers
21k views

Sum of odd numbers results in a square number [closed]

Recently I discovered by myself that sum of N sequential odd numbers will result into N2. Can anyone explain this to me? I want to know why not a proof. Explaining what a square is, is a Proof, ...
64
votes
15answers
13k views

f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

The question is about the function f(x) so that f(f(x))=exp (x)-1. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://scottaaronson....
3
votes
4answers
2k views

Positive vector bundles

In the case of a line bundle over M, positivity of such a bundle (one whose curvature form which is Kahler) gives rise to an embeddings of M into the projective space. Now I have in mind (more or ...
5
votes
5answers
947 views

A walk on a compact 2D surface embedded in 3-space that never returns home

At the risk of asking an uninformed question... Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once ...
2
votes
6answers
1k views

Computing zeta(k), for k odd, using Fourier coefficients

I'm not really sure what topics exactly this falls under, so I apologize if I've misclassified this question. There is a neat way of computing $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ using Fourier ...
13
votes
5answers
2k views

Is the wedge product of two harmonic forms harmonic?

Is the wedge product of two harmonic forms on a compact Riemannian manifold harmonic? I'm looking for a counter-example that the textbooks say exists. I would like to see a counter example that is on ...
70
votes
7answers
5k views

Roots of truncations of $ e^x - 1$

During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...
6
votes
3answers
1k views

Examples of birational equivalence of a variety and a hypersurface

There's an algebraic geometry theorem (I.4.9 in Hartshorne) that says: any variety of dimension r (over an algebraically closed field) is birationally equivalent to a hypersurface in projective space ...
3
votes
2answers
5k views

Proof of “if $a^2 + b^2 = c^2$ then $abc$ is divisible by 60”

Sorry if this question is too simple. I once read, on a number theory textbook - forget the title, in one of the problems list that all Pythagorean triplets when multiplied are divisible by 60. I ...
4
votes
1answer
594 views

Integrally closed factor rings and projective modules

I have a weird vision that comes from reading a paper by Raphael and Desrochers.. Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is ...
3
votes
5answers
736 views

Generating-functions: is there a relationship between a generating function and the corresponding squared generating function

Let's say we have a sequence $T(n)$ with the corresponding generating function $$A(t) = \sum_{n = 0}^\infty T(n) t^n$$ Is there some relationship between the two functions $A(t)$ and $A(t^2)$? And ...
5
votes
0answers
282 views

Real representations of G = those of Langlands dual and maps of a cylinder

There is a result about the real representations of a simple Lie group $G$ which is known as Soergel conjecture or Vogan duality. We'll focus on the formulation that for a fixed character of $G$ $\...
10
votes
2answers
1k views

Number of faithful representations of a finite group

Is it known how many faithful linear representations a finite group G has on a complex vector space of given dimension? What if G is abelian? I would even be interested in this special case: the ...
46
votes
21answers
16k views

Interesting applications of the Pigeon-hole Principle

I'm a little late in realizing it, but today is Pigeon-hole Day. Festivities include thinking about awesome applications of the Pigeon-hole Principle. So let's come up with some. As always with these ...
23
votes
2answers
2k views

Two functors from Grp to Grp?

It has been many years since I first read Categories for the Working Mathematician, but I still have a question about one of the first exercises. Question 5 in section 1.3 asks you to find two ...
-3
votes
2answers
470 views
10
votes
1answer
517 views

Level raising by prime powers

Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume ...
2
votes
3answers
838 views

Sobolev norms of eigenfunctions

Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
17
votes
6answers
12k views

Curriculum vitae: including grants you've applied for, not received (or not yet received).

I've heard from multiple sources now that one's CV should include grants you've applied for, even if you didn't receive them or won't find out if you've received them until after your CV goes out. I ...
8
votes
3answers
978 views

Singularity of sparse random matrices

The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
2
votes
1answer
269 views

k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?

If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...
36
votes
6answers
5k views

Why is Milnor K-theory not ad hoc?

When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is ...
4
votes
2answers
653 views

Godel's 1st incompleteness theorem - clarification.

This should be a trivial question for people who know Gödel's 1st incompleteness theorem. I quote the statement the theorem from wikipedia: "Any effectively generated theory capable of expressing ...
62
votes
9answers
16k views

Relating Category Theory to Programming Language Theory

I'm wondering what the relation of category theory to programming language theory is. I've been reading some books on category theory and topos theory, but if someone happens to know what the ...
33
votes
4answers
2k views

How should one approach tropical mathematics?

Let me preface this by saying that my background is pretty meagre (i.e. solid undergrad). However, a few months ago I came across this paper which presented an idea that struck me as really remarkable....
53
votes
4answers
52k views

Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
18
votes
3answers
1k views

Is there an L^p tauberian theorem?

From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
79
votes
10answers
8k views

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
5
votes
2answers
466 views

Complexity of determining if two graphs have same cycle matroid?

Consider the following question: Input: Two graphs G1 and G2 Question: Is the cycle matroid M(G1) isomorphic to the cycle matroid M(G2) What is the complexity of this question? It is well known ...
3
votes
2answers
1k views

Normal operators and it's spectrum in C*-algebras

If $A$ is a C*-algebra and $n$ is a normal element of $A$, then we have: (By Gelfand duality for example.) $\operatorname{spec}( |N| ) = | \operatorname{spec}(N) | := \left\{ | \lambda | ; \lambda \...
13
votes
7answers
5k views

Mathematical Physics? (Particularly computational)

I just saw a post like this one, but particularly for statistical mechanics, I thought I'd ask the question in general. Where does a mathematically trained person go to learn mathematical physics? By ...
41
votes
1answer
9k views

Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
9
votes
2answers
738 views

How to distinguish between natural and unnatural equivalences of categories

Some equivalences of categories are constructed by explicitly giving a pair of functors that are inverses up to isomorphism. For example, the equivalence between CRing^op and affine schemes is given ...
85
votes
29answers
16k views

Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...
1
vote
4answers
4k views

What are the components of a transpose operator from $\mathbb R^{n\times n}$ to $\mathbb R^{n\times n}$?

Say I'm working in the space of linear transformations from $\mathbb R^n$ to $\mathbb R^n$ and I've picked a basis so I can identify with any operator a component matrix in $\mathbb R^{n\times n}$. ...
39
votes
6answers
4k views

Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
4
votes
3answers
714 views

Why do branches of math vary in proof styles and what category are different branches in?

Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot ...
38
votes
9answers
6k views

Classification problem for non-compact manifolds

Background It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic). I'm also under the impression that there is ...
2
votes
2answers
186 views

Convexity Theorem of Hamiltonian actions - the connectedness part

Suppose we have a Hamiltonian action of a torus T=T^m=R^m/Z^m on a compact, connected symplectic manifold M. According to the convexity theorem, we know every fiber of the momentum map \mu: M--->R^m ...
11
votes
3answers
3k views

Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...

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