8
votes
1answer
537 views

Do all combinatorially distinct fundamental polygons correspond to surfaces?

The topology of a closed surface can be constructed by identifying edges of a fundamental polygon of an even number $2n$ of edges. Labeling the edges and using $\pm 1$ exponents to indicate direction, ...
7
votes
1answer
391 views

K-theory of the h-cobordism category

I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...
73
votes
14answers
7k views

Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
23
votes
1answer
760 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
33
votes
2answers
1k views

Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher ...
42
votes
10answers
5k views

Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math. I've found that in talking to professional physicists and ...
25
votes
10answers
4k views

Value of “of course” in the mathematical literature

I've been thinking about the value of writing "of course" in mathematical papers (or its variants such as "obviously" etc.). In particular, my current train of thought is, if something is obvious, ...
7
votes
1answer
299 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
2
votes
2answers
214 views

Graded-irreducible ideals are irreducible?

One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals? Let $I$ be a graded ideal in a polynomial ring over a ...
24
votes
7answers
2k views

Applications of Frobenius theorem and conjecture

A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of ...
5
votes
2answers
522 views

pullback of theta divisor

Let $C$ be a smooth curve and $J$ its Jacobian. Let $p$ be a point on $C$ and $j: C \to J$ be the map $x \mapsto x-p$. Let $\theta$ be the Theta divisor on $J$, i.e. the locus $\{ x_1 + \cdots + ...
92
votes
3answers
9k views

Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open. I would think that the question of its convergence is really ...
14
votes
7answers
3k views

Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
7
votes
1answer
162 views

Do coherent toposes descend along open surjection?

Let $f:\mathcal{L} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{T}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...
21
votes
6answers
2k views

undecidable sentences of first-order arithmetic whose truth values are unknown

Godel's undecidable sentences in first-order arithmetic were guaranteed to be true, by construction. But are there examples of specific sentences known to be undecidable in first-order arithmetic ...
38
votes
10answers
7k views

Definition of “simplicial complex”

When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition". However, ...
137
votes
3answers
7k views

A Game on Noetherian Rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...
2
votes
0answers
422 views

Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation. Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
8
votes
1answer
240 views

Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
30
votes
7answers
3k views

formal power series convergence

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed, below, identified by the symbol $g$ because I am not entirely sure ...
16
votes
8answers
8k views

Does the exponential function have a square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function h(x) such that h(h(x))=ex? Is it unique? Is it analytic? Related question: Is there an invertible ...
5
votes
0answers
874 views

Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
34
votes
6answers
4k views

Is the Mendeleev table explained in quantum mechanics?

Does anybody know if there exists a mathematical explanation of Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the authors ...
17
votes
4answers
2k views

Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...
24
votes
3answers
2k views

What manifolds are bounded by RP^odd?

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 ...
52
votes
6answers
5k views

Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes: Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
15
votes
5answers
2k views

Why does the (S2) property of a ring correspond to the Hartogs phenomenon?

Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this ...
38
votes
7answers
6k views

Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders ...
24
votes
6answers
3k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find ...
7
votes
1answer
436 views

A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds, \begin{equation*} ...
7
votes
6answers
4k views

Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
37
votes
2answers
3k views

Alternating sum of square roots of binomial coefficients

Let $$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
18
votes
4answers
1k views

Fairest way to choose gifts

Suppose that a parent brings home from a trip $2n$ gifts of roughly equal value for his/her two children. The children get to choose one at a time which gifts they want. What is the fairest way to do ...
0
votes
2answers
843 views

Standard model of ZFC

Is ZFC+Con(ZFC) powerful enough to show there isn't any standard model of ZFC? What you think about it?
0
votes
2answers
982 views

Existence of solution to quasilinear parabolic PDEs

Hello. I want to prove the existence of a weak solution to: Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that $$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$ ...
8
votes
2answers
960 views

Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...
14
votes
2answers
608 views

Normal subgroups of finite index in free groups

Hi all, This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups ...
7
votes
12answers
3k views

Graphs with fractal properties?

For the purposes of a research project, I am wondering if there are any resources on graphs with fractal properties, by which I mean self-similarity in particular. For instance, imagine a graph where ...
10
votes
4answers
1k views

Elements of infinite order in a profinite group

Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general? A start for (A): we can ask the same question ...
13
votes
7answers
5k views

Nash embedding theorem for 2D manifolds

The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$. What can we say in the special case of 2-manifolds? For example, can ...
31
votes
4answers
5k views

What is Chern-Simons theory?

What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics. ...
9
votes
1answer
773 views

Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum. In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO). On the other hand, MU and ...
10
votes
3answers
2k views

Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces. In this connection, I have read Chapter 11 of the book Primer of mapping ...
2
votes
1answer
374 views

From Lebesgue Integral to Stieltjes Integral, and integration by parts

Let $X$ be a real random variable with c.d.f function $F$. Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite). What additional ...
1
vote
1answer
121 views

contact metric structure on squashed spheres

My goal to write down an explicit (and simplest) contact metric structure on squashed $S_\omega^{2n + 1}$ defined as \begin{equation} S_\omega ^{2n + 1} = \left\{ {\left( {{z_i}} \right) \in ...
17
votes
4answers
3k views

Intuitive explanation of Burnside's Lemma

Burnside's Lemma states that, given a set $X$ acted on by a group $G$, $$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$ where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of ...
10
votes
5answers
2k views

Simple/efficient representation of Stirling numbers of the first kind

Stirling numbers of the second kind can be expressed by means of a simple hypergeometric (considering $n$ fixed) sum $$S_2(n,k) = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n. \qquad (1)$$ ...
-3
votes
2answers
886 views

Should science authors discourage / boycott the recent push for author IDs [closed]

In recent years, several organizations (publishers, arXiv, universities) started pushing for systems of a reliable author identification, gaining considerable traction with the recent launch of ORCID. ...
7
votes
2answers
495 views

Relating curvature and torsion of a connection to those of a curve

I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion. I know that an affine connection $A$ on an $n$-dimensional manifold ...
16
votes
1answer
1k views

Manifolds with two coordinate charts

What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$? And is it true that ...

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