All Questions

1
vote
2answers
2k views

Number of irreducible representations [closed]

According to Wikipedia: If G is a finite group and K is the complex number field, the regular representation is a direct sum of irreducible representations, in number at least the number of conjugacy ...
11
votes
3answers
751 views

How much “Morse theory” can be accomplished given only a continuous transformation of a space?

If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...
2
votes
1answer
7k views

Number of Shortest paths problem

Hey, Is countinng the number of shortest paths in a weighted directed acyclic graph with nonnegative weights #P-complete? If so, is there a proof I can read somewhere? Thanks
11
votes
3answers
1k views

level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
21
votes
8answers
2k views

Can we disallow finite choice?

When people work with infinite sets, there are some who (with good reason) don't like to use the Axiom of Choice. This is defensible, since the axiom is independent of the other axioms of ZF set ...
34
votes
3answers
2k views

Are there pairs of highly connected finite CW-complexes with the same homotopy groups?

Fix an integer n. Can you find two finite CW-complexes X and Y which * are both n connected, * are not homotopy equivalent, yet * $\pi_q X \approx \pi_q Y$ for all $q$. In Are there two non-...
16
votes
3answers
1k views

The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points. One high dimensional extension ...
12
votes
3answers
1k views

Classifying triangulated structures on a graded category

I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me ...
26
votes
10answers
6k views

When to pick a basis?

Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the ...
12
votes
3answers
693 views

Are supervector spaces the representations of a Hopf algebra?

Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...
2
votes
1answer
246 views

Hausdorff Derived Series

There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
8
votes
1answer
1k views

Explanation for Satake correspondence

Some time ago I was told there's an interesting classical Satake correspondence which I will write as $$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \...
5
votes
3answers
585 views

Regularity of sparse Fourier transforms

Suppose $F$ has discrete Fourier transform $(a_n)$ where $a_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a_n=1/k$ (or $a_n=1/k^2$ if you want: I'm happy with anything polynomial). What ...
15
votes
8answers
3k views

Hironaka desingularisation theorem — new proofs in literature?

I'm wondering what the landscape looks like for proofs of Hironaka's desingularisation theorem. Are there many proofs in the literature? Is there a commonly accepted simplest bare-knuckle proof ...
0
votes
3answers
420 views

Explanation and Definition of Iwahori order

Can anyone explain what Iwahori order is? All I know is that it is mentioned here.
5
votes
4answers
1k views

Inverting Ramanujan's partition function, p(N)

Would someone be so kind as to enlighten me as to whether the integer partition function, p(N), can be (or has been) inverted and where the inversion is recorded? I'm trying to avoid reinventing the ...
15
votes
4answers
12k views

On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + …

It is well-known that A: The series of the reciprocals of the primes diverges My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers. Property A ...
8
votes
5answers
2k views

Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring

Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
13
votes
5answers
3k views

When are dual modules free?

Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes_A K$ is a finite dimensional vector space $V$. Let $T^*$ be ...
11
votes
9answers
2k views

Is there a non self-referencing non-computable function?

I've seen in college that some functions are not computable. The proof for that was the case of Halt(x,y) function. The thing is, the proof used a very artificial (IMHO) case which is evaluating ...
9
votes
1answer
428 views

Can an algebraic space fail to have a unviersal map to a scheme?

Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...
7
votes
1answer
833 views

Valuative criterion for properness

Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any $K$-...
7
votes
1answer
721 views

Analogue of Sperner's lemma for Lefschetz theorem?

Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question. One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for ...
12
votes
7answers
1k views

Upper bound on the area of a midpoint pentagon?

Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to ...
19
votes
5answers
4k views

Pacing for learning new material [closed]

I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best ...
19
votes
3answers
3k views

Subgroups of free abelian groups are free: a topological proof?

There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which ...
18
votes
3answers
3k views

When is an algebraic space a scheme?

Sometimes general theory is "good" at showing that a functor is representable by an algebraic spaces (e.g., Hilbert functors, Picard functors, coarse moduli spaces, etc). What sort of general ...
18
votes
5answers
5k views

Maps to projective space determined by a line bundle

The following should be pretty standard for any algebraic geometer. Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...
22
votes
2answers
1k views

Is there an infinity × infinity lemma for abelian categories?

Many people know that there is a (3×3) nine lemma in category theory. There is also apparently a sixteen lemma, as used in a paper on the arXiv (see page 24). There might be a twenty-five lemma, as ...
35
votes
3answers
6k views

What is the “intuition” behind “brave new algebra”?

Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
6
votes
1answer
787 views

Uniformization in algebraic/arithmetic geometry?

Jonah's question makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. this article by Faltings seems to be about that, the Shimura-Taniyama statement too, Mochizuki ...
2
votes
4answers
513 views

a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...
33
votes
5answers
7k views

Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...
4
votes
2answers
1k views

Moment map for toric actions — online references?

Consider a toric variety, defined as a (normal?) complex projective variety $X$ together with an algebraic action of $(\mathbb C^*)^n$ with finitely many orbits. Now we have two "real symplectic" ...
1
vote
2answers
3k views

Elliptical rotation matrix [closed]

We can rotate a point 'circularly' about an arbitrary axis: the equation is here, but this site doesn't trust me enough yet to post an image., But as we walk theta 0 -> 2PI this takes the point ...
1
vote
4answers
354 views

Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?

Does this even make sense what I translated into english? PS. I am probably gonna delete this question eventually
5
votes
2answers
3k views

Something like mathoverflow in other sciences [closed]

Are the sites similar to mathoverflow in other sciences related to mathematics? statistics, computer science, physics, economics, etc? Let me explain what I mean by "similar": those are sites devoted ...
10
votes
3answers
595 views

What happens to Newtonian systems as the mass vanishes?

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one. Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $...
15
votes
5answers
2k views

Can we count isogeny classes of abelian varieties?

Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
44
votes
12answers
7k views

Cures for mathematician's block (as in writer's block) [closed]

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening ...
14
votes
2answers
971 views

Exotic spheres and stable homotopy in all large dimensions?

Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the Kervaire-...
48
votes
4answers
4k views

Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...
10
votes
5answers
653 views

Assumptions on the category C for sheafification of C-valued presheaves

For any category C and topological space X we have the notion of a C-valued presheaf on X. What assumptions must be made about C in order that we have the notion of such a presheaf being a 'sheaf'? I ...
45
votes
7answers
5k views

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello, I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ : For example : 1) Are all ...
3
votes
13answers
3k views

Definition of elementary number theory

It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." I have two closely related questions. Is my ...
9
votes
2answers
3k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
6
votes
5answers
457 views

What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

Background Lagrangian mechanics on $\mathbb R^n$ is usually defined by picking a Lagrangian function $L: {\rm T}\mathbb R^n \to \mathbb R$, where ${\rm T}\mathbb R^n = \mathbb R^{2n}$ is the tangent ...
17
votes
4answers
2k views

How many of the true sentences are provable?

Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or ...
6
votes
11answers
3k views

Is there a theorem that says that there is always more than one way to “continue a finite sequence”? [closed]

I have come across a bit of folklore(?) which goes something like "given any finite sequence of numbers, there is more than one 'valid' way of continuing the sequence". For example see here. I would ...
19
votes
6answers
1k views

Elementary solutions to f(z+1)-f(z)=g(z) in entire functions

Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this is apparently 'yes', ...

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