All Questions
17,349
questions
37
votes
4
answers
2k
views
How many morphisms from 1 to 1+1 can there be?
Here is an interesting question raised by Alice Rhyl.
Let $C$ be a category with a terminal object $1$ and finite coproducts. How many different morphisms $f : 1 \to 1 + 1$ can there be?
There are ...
- 21.5k
37
votes
4
answers
6k
views
For which $n$ is there only one group of order $n$?
Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts:
If $n$ is not squarefree, then there are multiple abelian groups of order $n$.
If $n \geq 4$ is even,...
- 1,806
37
votes
5
answers
3k
views
Usefulness of using TQFTs
What is a topological feature, that a (some) TQFT (e.g. in 3 or 4 dim) sees but homology/cohomology/homotopy groups don't? Or: what is an example where using classical theories is hard, but using a ...
- 843
37
votes
11
answers
8k
views
"Must read" papers in numerical analysis
In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis.
In Trefethen's words, ... this course ...
37
votes
15
answers
12k
views
Geometric imagination of differential forms
In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
- 2,031
36
votes
3
answers
2k
views
The roots of unity in a tensor product of commutative rings
For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
- 4,435
36
votes
7
answers
14k
views
Changing Careers: Becoming a Professional Mathematician
First, I apologize if this question is too soft or doesn't comport precisely with what is considered a good question, but I know of no other place to ask it.
A little background: I obtained an ...
36
votes
1
answer
3k
views
Artin reciprocity $\implies $ Cubic reciprocity
I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...
- 1,167
36
votes
3
answers
6k
views
In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...
- 4,792
36
votes
2
answers
4k
views
Schubert calculus, as lowbrow as possible
Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...
- 1,510
36
votes
2
answers
3k
views
Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.6k
36
votes
5
answers
4k
views
What is the general geometric interpretation of modules in algebraic geometry?
Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question.
I understand that the basic philosophy ...
- 2,007
36
votes
4
answers
5k
views
Construction of the Stiefel-Whitney and Chern Classes
I've seen two constructions of these characteristic classes. The first comes from Milnor and Stasheff's book and involves the Thom isomorphism and (at least for me) the rather mysterious Steenrod ...
- 3,938
36
votes
3
answers
4k
views
the following inequality is true，but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
- 363
36
votes
6
answers
2k
views
When are some products of gamma functions algebraic numbers?
I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...
- 13.9k
36
votes
7
answers
2k
views
Informal online seminars or reading groups via videoconferencing?
Does the following exist, and if not, does anyone besides me wish it did? A web site where a mathematician (say) could find other mathematicians who want to study the same book or paper, and arrange ...
36
votes
3
answers
5k
views
$\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$
When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this:
$$\prod_{n=1}^\infty n^{\mu(n)}=\frac{1}{4 \pi ^2}$$
Note: this is the reciprocal of (3) zeta-...
- 1,043
36
votes
0
answers
1k
views
Functor that maps to both $KO^n$ and $KO^{-n}$
(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...
- 42.5k
36
votes
3
answers
4k
views
What are D-branes, really?
In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
- 1,405
36
votes
2
answers
3k
views
Does anyone want a pretty Maass form?
A few months ago, I was curious about some properties of Maass cusp forms, of nonabelian arithmetic origin. As a result, I went through a somewhat predictable process of finding a totally real $A_4$ ...
36
votes
3
answers
5k
views
What do higher Chow groups mean?
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
- 12k
36
votes
5
answers
4k
views
Advice for researchers outside academia
Perhaps some personal background is relevant to this question. A couple of years ago, I graduated with a master's degree in Applied Mathematics from a good Dutch university. Even though I obtained ...
36
votes
2
answers
6k
views
Why do primes dislike dividing the sum of all the preceding primes?
I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
- 4,638
36
votes
3
answers
5k
views
Mazur's unpublished manuscript on primes and knots?
The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur.
I'm not absolutely sure this is the one I mean, but in his paper, ...
- 17.5k
36
votes
3
answers
8k
views
Tools for the Langlands Program?
Hi,
I know this might be a bit vague, but I was wondering what are the hypothetical tools necessary to solve the Langlands conjectures (the original statments or the "geometic" analogue). What I mean ...
- 361
36
votes
1
answer
3k
views
Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
- 61.5k
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...
36
votes
6
answers
3k
views
Number of real roots of 0,1 polynomial
$0,1$ polynomial has coefficients from $\{0,1\}$.
I investigate the number of roots in such polynomials.
We are talking about real roots, and multiples are counted only once.
It was found numerically ...
- 462
36
votes
3
answers
3k
views
When is a closed differential form harmonic relative to some metric?
Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic,...
- 585
36
votes
2
answers
3k
views
The coupon collector's earworm
[EDITED mostly to report on the answer by Kevin Costello
(and to improve the gp code at the end)]
I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
...
- 77.5k
36
votes
6
answers
3k
views
Distinct well-orderings of the same set
An easy consequence of the Erdős-Dushnik-Miller theorem $\kappa\to(\kappa,\omega)^2$ is the following, that will denote $(*)$ (it appears as an exercise in Kunen's book, it was probably mentioned ...
- 32.2k
36
votes
5
answers
6k
views
Tips on cohomology for number theory
I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.
Do people just remember all the rules and ...
36
votes
6
answers
4k
views
Taylor's theorem and the symmetric group
Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: ...
- 12k
36
votes
5
answers
6k
views
What is the equivariant cohomology of a group acting on itself by conjugation?
This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant ...
- 13.2k
36
votes
14
answers
5k
views
What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
- 32.8k
35
votes
2
answers
7k
views
Why polarization of abelian varieties?
Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...
- 2,199
35
votes
1
answer
3k
views
Is there a common genesis for ADE classifications?
Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one ...
- 28.8k
35
votes
6
answers
9k
views
Do convolution and multiplication satisfy any nontrivial algebraic identities?
For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication -
$$(f\cdot g)(x) = f(x)g(x),$$
and convolution -
$$(f*...
- 5,932
35
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 6,787
35
votes
4
answers
2k
views
Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...
- 2,517
35
votes
3
answers
6k
views
What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
- 10k
35
votes
6
answers
4k
views
Why Drinfel'd-Jimbo-type quantum groups?
Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
- 12.7k
35
votes
2
answers
4k
views
Good covers of manifolds
It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a good cover, i.e., a locally finite cover by open balls such that all nonempty intersections of the ...
- 31k
35
votes
1
answer
3k
views
"The Two Sheriffs" puzzle
This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler.
Two sheriffs in neighboring towns are on the track of a killer, in a
case involving eight ...
- 11.9k
35
votes
14
answers
4k
views
Where have you used computer programming in your career as an (applied/pure) mathematician?
For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have.
...
35
votes
10
answers
8k
views
Some questions about scalar curvature
Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
...
- 28.8k
35
votes
7
answers
4k
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 ...
- 2,197
35
votes
6
answers
7k
views
The current status of the Birch & Swinnerton-Dyer Conjecture
[Une traduction française suit la version anglaise.]
The question is only about elliptic curves $E$ over $\mathbb{Q}$ and concerns only the aspect
(order of vanishing of $L(E,s)$ at $s=1$)$\ =\ $(...
- 18.5k
35
votes
1
answer
3k
views
Proving the irrationality of $\pi e$ and $\pi / e$
Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
- 1,459
35
votes
7
answers
11k
views
Fraktur symbols for Lie algebras
Does anyone know when and why the Fraktur script was introduced for Lie and other algebras—$\mathfrak{g}$, $\mathfrak{gl}_n$, $X/\mathfrak{g}$,
$\mathfrak{g}\oplus\mathfrak{g}$, $\mathfrak{su}$, ...
- 149k