All Questions

20
votes
13answers
6k views

Category theory sans (much) motivation?

So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
3
votes
1answer
335 views

Weil-Châtelet group

Sorry if this is obvious. I'd like to understand why the map WC(E/Q) -> H^1(Gal(Q/Q), E(Q)) is bijective. Thanks.
9
votes
4answers
1k views

Does Cantor-Bernstein hold for classes?

In Bonn, we've been have a discussion on the topic in the title: Suppose that A and B is are classes and that there are injections from A to B and fom B to A. Does it follow that there is a ...
12
votes
1answer
949 views

What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?

Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...
49
votes
22answers
12k views

What's a groupoid? What's a good example of a groupoid? [closed]

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
10
votes
4answers
786 views

easy(?) probability/diff eq. question

I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
14
votes
8answers
2k views

Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
5
votes
2answers
284 views

Smooth immersion(?) of graphs into the plane

Sorry if the terminology's wrong, I don't know differential topology. Also, this is more of a brain-teaser than a bona fide research question, but it's hopefully a "real mathematician"-level brain-...
13
votes
1answer
661 views

Commutativity in K-theory and cohomology

The Chern classes give a map $f : BU \to \prod_n K(\mathbb{Z},2n)$, which is a rational equivalence. However, it is not an equivalence over $\mathbb{Z}$ because the cohomology of $BU$ is just a ...
155
votes
80answers
124k views

Do good math jokes exist? [closed]

Have a good joke? Share. I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)
8
votes
6answers
2k views

What is an example of a topological space that is not homotopy equivalent to a CW-complex?

It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes: "The homotopy category of CW complexes is, in the opinion of some experts, the best if not ...
14
votes
4answers
2k views

How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
3
votes
3answers
920 views

Boolean network as a gauge field

Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two ...
5
votes
4answers
996 views

Motivation for coherence axioms

The pentagon and hexagon axioms in the definition of a symmetric monoidal category are one example that I was thinking of here; the axioms of a weak 2-category are another. I understand that it can ...
45
votes
1answer
7k views

Order of an automorphism of a finite group

Let G be a finite group of order n. Must every automorphism of G have order less than n? (David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil ...
6
votes
5answers
3k views

Definition of infinite permutations

I've been trying to find a definition of an infinite permutation on-line without much success. Does there exist a canonical definition or are there various ways one might go about defining this? The ...
-1
votes
1answer
261 views

Change of basis with Multilinear fucntion [closed]

Take a multi-linear function(or functional) M that takes m arguments V1…Vm, each with a dimension n. Consider only the case where m=n. Let there be a change of basis performed on the arguments(V1...Vm)...
73
votes
7answers
6k views

When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...
9
votes
4answers
764 views

What m minimizes E(|m-X|^3) for a random variable X?

Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x. A couple weeks ago in a technical ...
18
votes
4answers
1k views

Splitting Pythagorean triples

Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would be least surprised ...
35
votes
12answers
16k views

Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do $$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why ...
10
votes
2answers
985 views

What is the size of the category of finite dimensional F_q vector spaces?

The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
6
votes
4answers
584 views

decidability of group homomorphism existence

Fix a finitely-presented group $G$ with distinguished non-identity element $g$. For any finitely-presented group $H$ with element $h$, is it decidable whether there is a homomorphism $h: G \...
3
votes
4answers
1k views

Decomposing a 1-d signal into arbitary basis functions

Hi all, The short-time fourier transform decomposes a signal window into a sin/cosine series. How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead ...
6
votes
2answers
866 views

Explicit Direct Summands in the Decomposition Theorem

Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
19
votes
2answers
3k views

“Fermat's last theorem” and anabelian geometry?

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
2
votes
2answers
339 views

Limit of sequence involving gamma functions

Let G be the gamma function, and b be a constant in (-2,inf). Let H(n, i) = G(i+1+b) * G(n-i+1+b) / [G(i+1) * G(n-i+1)] for integers n > i > 0. Let S(n) = \sum_{i=1}^{i=n-1} H(n, i). Let x_ n = H(...
2
votes
2answers
7k views

Latex Template for a Popular Math Journal [closed]

Can anyone offer a Latex template for a popular mathematics journal? It is easy to prepare a template for a technical journal with simple page layout but what I am looking for is something like the ...
13
votes
7answers
2k views

Intro to automatic theorem proving / logical foundations?

Is there any web-based course or materials about logic / automatic theorem proving? (I checked MIT's OpenCourseWare and I only found a vaguely related AI course)
9
votes
4answers
518 views

What is the right way to think about / represent general tilings?

For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
22
votes
4answers
3k views

Motivation/interpretation for Quillen's Q-construction?

This question has been on my mind for a while. As I understand it, the Q-construction was the first definition for higher algebraic K-theory. Some details can be found here. http://en.wikipedia.org/...
7
votes
1answer
639 views

Example where you *need* non-DVRs in the valuative criteria

The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it ...
50
votes
9answers
5k views

Is every finite group a group of “symmetries”?

I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
7
votes
4answers
983 views

For which hypersurfaces in projective space does the complement admit an algebraic group structure?

For example, if $H$ is a hyperplane, then $\mathbb{P}^n - H = \mathbb{A}^n$, which is a vector space. If $n = m^2 - 1$, then we can regard $\mathbb{A}^{n+1}$ as the space of $m \times m$ matrices and ...
10
votes
6answers
4k views

Applications and Natural Occurrences of Prime Numbers

I'm fascinated by prime numbers, and over the years, I've found multiple applications and natural occurrences for them. But can anyone suggest some alternatives that aren't in my list? Applications ...
10
votes
2answers
2k views

algebraic K-theory and tensor products

Algebraic K-theory defines a functor K taking commutative rings to E_\infty ring spectra. I'm interested in which pushouts (tensor/smash products) K preserves. For example, if R is a regular ...
14
votes
7answers
3k views

What is lambda calculus related to?

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based. I was wondering if anyone had a suggestion ...
9
votes
5answers
2k views

Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in http://...
33
votes
10answers
16k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
9
votes
3answers
999 views

Does Ribet's level lowering theorem hold for prime powers?

I often use the following theorem (that one can state more generally) in my research. Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or ...
13
votes
5answers
2k views

What is the Hilbert class field of a cyclotomic field?

In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...
6
votes
3answers
884 views

Looking for cubic, bipartite graphs with girth at least six and no cycles of length 8.

Aside from the Desargues graph, are there nice (at least vertex-transitive), small (say, less than 60 vertices), cubic, bipartite graphs with girth at least 6 and no 8-cycles? (or, even better, no ...
9
votes
4answers
3k views

Fourier transform of exp(-||x||_p): more general question

David Corfield asked the following questions yesterday: Is the n-dimensional Fourier transform of exp(-||x||) always non-negative, where ||.|| is the Euclidean norm on R^n? What is its support? I ...
34
votes
5answers
4k views

Analogue to covering space for higher homotopy groups?

The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched ...
84
votes
12answers
24k views

Where are mathematics jobs advertised if not on mathjobs (e.g. in Europe and elsewhere)?

My impression is that in the US, there is a canonical place for finding math jobs, namely mathjobs.org. For those of us who live and apply for jobs elsewhere, life is more complicated, and searching ...
3
votes
2answers
368 views

Legendrian homotopy of curves in a contact structure?

I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...
16
votes
5answers
2k views

How Does One Find the “Loneliest Person on the Planet”?

I'm looking for the algorithm that efficiently locates the "Loneliest Person on the Planet", where "loneliest" is defined as: Maximum minimum distance to another person -- that is, the person for ...
5
votes
2answers
736 views

Operator Valued Weights

One of the basic tools in subfactors is the conditional expectation. If $N\subset M$ is a $II_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional ...
5
votes
1answer
482 views

a general theory of configurations?

Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
0
votes
2answers
2k views

Friedberg, Insel, and Spence Linear Algebra example

In the chapter 6.4 on normal and self-adjoint operators, there is an example of an infinite dimensional inner product space H that has a normal operator but that has no eigenvectors. The space is the ...

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