All Questions

6
votes
4answers
586 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
877 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
519 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
661 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
992 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://...
35
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
1k 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
890 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 ...
35
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 ...
85
votes
12answers
25k 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
743 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
484 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 ...
1
vote
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 ...
-3
votes
2answers
615 views

Cycle Length of the Positive Powers of Two Mod Powers of Ten [closed]

I want to prove that the positive powers of two, mod 10m, cycle with period 4*5m-1. It's simple to prove that the powers of FIVE cycle with this period (2 is a primitive root mod powers of five), but ...
65
votes
12answers
7k views

Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
13
votes
6answers
2k views

Why the search for ever larger primes?

I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the ...
35
votes
11answers
12k views

Resources for learning practical category theory

I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but ...
6
votes
4answers
2k views

Integer division: the length of the repetitive sequence after the decimal point

When dividing two integers, there may be an infinite sequence of digits after the decimal point (e.g. in the cases of 1/3, 1/7 etc). As far as I know, if the two numbers divided are integers, this ...
22
votes
8answers
2k views

Points and lines in the plane

Does a positive real number $k\geq1$ exist such that for every finite set $P$ of points in the plane (with the property that no three points of $P$ lie on a common line and $|P|\geq3$), one can choose ...
7
votes
3answers
2k views

Why is the Euler characteristic of powers of a line bundle a polynomial in the power?

Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor ...
166
votes
42answers
51k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
11
votes
4answers
2k views

Short Introduction to Planar Algebras

Are there any good short expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.
5
votes
2answers
952 views

Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
25
votes
7answers
3k views

Etale covers of the affine line

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
11
votes
0answers
751 views

Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
1
vote
4answers
2k views

Closest grid square to a point in spherical coordinates

I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on ...
39
votes
1answer
16k views

What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
3
votes
3answers
1k views

Conjugation in SU(2)

For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
19
votes
3answers
2k views

Is any representation of a finite group defined over the algebraic integers?

Apologies in advance if this is obvious.
1
vote
8answers
2k views

The core question of topology

As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces. To answer this question, one defines various properties of a space such as ...
5
votes
1answer
438 views

Do homotopy pullbacks commute with homotopy orbits (in spaces)?

Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{...
5
votes
2answers
390 views

Algorithms for semistable reduction of families of curves

This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a ...
15
votes
4answers
2k views

Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? : Is it known that there are infinitely many primes p for which ...
7
votes
2answers
662 views

Is there a version of the valuative criteria for separateness/properness for varieties?

What I had in mind was something like the following: X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way. Is there a good reason why ...
4
votes
2answers
422 views

Embedding abelian categories to have enough projectives

Is it true that the pro-objects of an abelian category form a category with enough projectives? In general, given an abelian category A, is there a canonical way to embed it a bigger abelian ...

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