Frequent Questions
18,054 questions
43
votes
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answers
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Approaches to Riemann hypothesis using methods outside number theory [closed]
Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...
43
votes
4
answers
6k
views
Why the Dold-Thom theorem?
Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
- 17.5k
42
votes
4
answers
8k
views
Tannakian Formalism
The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...
- 5,548
41
votes
5
answers
11k
views
Mathematically mature way to think about Mayer–Vietoris
This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
- 6,348
40
votes
5
answers
6k
views
Probabilities in a riddle involving axiom of choice
The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle:
We assume ...
- 1,341
38
votes
3
answers
4k
views
What is the point of pointwise Kan extensions?
Recall that a Kan extension is called pointwise if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors.
I have seen pointwise Kan ...
- 64k
38
votes
5
answers
4k
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When factors may be cancelled in homeomorphic products?
It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{...
- 109k
38
votes
3
answers
3k
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How do you define (infinity,1) categories in Homotopy Type Theory?
One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
- 28.1k
38
votes
5
answers
10k
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Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
37
votes
6
answers
6k
views
Who needs Replacement anyway?
The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
- 35.5k
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
36
votes
1
answer
3k
views
Whence “homomorphism” and “homomorphic”?
Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?
“Homomorphic” (and “homomorphism” as “property of being ...
- 31.5k
35
votes
6
answers
3k
views
On the universal property of the completion of an ordered field
I have been trying to write up some notes on completion of ordered fields, ideally in the general case (i.e., not just completing $\mathbb{Q}$ to get $\mathbb{R}$ but considering the completion via ...
- 65.4k
35
votes
5
answers
4k
views
Cliques, Paley graphs and quadratic residues
A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...
- 703
34
votes
8
answers
8k
views
Arithmetic fixed point theorem
I want to understand the idea of the proof of the artihmetic fixed point theorem. The theorem is crucial in the proof of Gödel's first Incompletness theorem.
First some notation: We work in $NT$, the ...
- 63.1k
33
votes
1
answer
1k
views
Nilpotence of the stable Hopf map via framed cobordism
The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
- 28.1k
32
votes
3
answers
7k
views
Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, asserting that ...
- 118k
31
votes
11
answers
13k
views
Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
30
votes
3
answers
3k
views
Modular forms of fractional weight
Modular forms of integral weight are prominent in number theory.
Furthermore, there are $\theta$-functions and the $\eta$-function, having weight 1/2,
which also have a rich theory.
But I have never ...
- 2,810
30
votes
3
answers
3k
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Can there be an embedding j:V → L, from the set-theoretic universe V to the constructible universe L, when V ≠ L?
Main Question. Can there be an embedding $j:V\to L$ of the
set-theoretic universe $V$ to the constructible universe $L$, if
$V\neq L$?
By embedding here, I mean merely a proper class isomorphism from
$...
- 236k
30
votes
3
answers
5k
views
When is a classification problem "wild"?
I hope someone can point me to a quick definition of the following terminology.
I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
- 33.3k
29
votes
6
answers
5k
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Infinitely many primes of the form $2^n+c$ as $n$ varies?
At the time of writing, question 5191 is closed with the accusation of homework. But I don't have a clue about what is going on in that question (other than part 3) [Edit: Anton's comments at 5191 ...
- 41.4k
29
votes
4
answers
5k
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Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
- 291
29
votes
6
answers
10k
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how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
- 2,600
28
votes
2
answers
7k
views
Large cardinal axioms and Grothendieck universes
A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
- 23.5k
28
votes
3
answers
6k
views
Motivation of Virasoro algebra
I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with ...
- 1,663
28
votes
1
answer
2k
views
Have finite doubly transitive groups been classified?
I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas ...
- 6,397
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
26
votes
6
answers
8k
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prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
26
votes
5
answers
10k
views
Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?
It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
- 14k
26
votes
2
answers
9k
views
Maximal ideals in the ring of continuous real-valued functions on ℝ
For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
- 6,734
25
votes
9
answers
6k
views
Function with range equal to whole reals on every open set
There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.
I want to generalize this in a way to get a function ...
- 2,821
24
votes
7
answers
16k
views
Expected determinant of a random NxN matrix
What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
- 251
24
votes
3
answers
1k
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Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
- 343
24
votes
5
answers
8k
views
totally disconnected and zero-dimensional spaces
When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...
- 3,605
23
votes
4
answers
3k
views
Non-vanishing of group cohomology in sufficiently high degree
Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as $ K(BG)$, gives bunch of corollaries of ...
- 1,003
23
votes
3
answers
6k
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Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 505
23
votes
4
answers
9k
views
Integer points of an elliptic curve
Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. ...
- 261
23
votes
2
answers
2k
views
Orbit structures of conjugacy class set and irreducible representation set under automorphism group
let G be a finite group. Suppose C is the set of conjugacy classes of G and R is the set of (equivalence classes of) irreducible representations of G over the complex numbers.
The automorphism group ...
- 7,320
22
votes
2
answers
1k
views
A closed form for an integral expressed as a finite series of $\zeta(2k+1)$, $\pi^m$ and a rational?
In this paper the following beautiful integral expression for $\zeta(3)$ is derived:
$$\zeta(3)=\frac{1}{7}\,\int_0^{\pi} x\,(\pi-x)\csc(x)\, dx$$
In a comment at the end of this question, I ...
- 4,169
22
votes
4
answers
3k
views
Can you determine whether a graph is the 1-skeleton of a polytope?
How do I test whether a given undirected graph is the 1-skeleton of a polytope?
How can I tell the dimension of a given 1-skeleton?
- 9,720
21
votes
4
answers
5k
views
Extending vector bundles on a given open subscheme, reprise
In this question, Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist.
Sasha's answer suggests that extensions of vector bundles don't always exist.
More ...
- 21k
21
votes
4
answers
2k
views
Squarefree parts of Mersenne numbers
The $n$-th Mersenne number is $M_n=2^n-1$. Write $M_n=a_n b_n^2$ where $a_n$ is positive and squarefree.
Question 1: What lower bound can be proved for $a_n$?
Let $A$ be the set of all possible $...
- 3,142
21
votes
3
answers
7k
views
What are the current breakthroughs of Geometric Complexity Theory?
I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods.
That program seems ...
21
votes
5
answers
3k
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How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?
Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
- 24.9k
21
votes
6
answers
3k
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Where in ordinary math do we need unbounded separation and replacement?
[I have updated the question after initial comments in the hope of clarifying it.]
I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as ...
20
votes
11
answers
16k
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What to do with antique math books?
My grandfather had a PhD in math. When he died, he left a lot of math textbooks, which I took. These include things like Van der Waerden's 2-volume algebra set from the 1970s,
"Studies in Global ...
19
votes
3
answers
5k
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What determines a model structure?
It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
cofibrations and weak ...
- 2,782
19
votes
2
answers
1k
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Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 149k
19
votes
4
answers
3k
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Equations satisfied by the Riemann curvature tensor
It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
\end{...
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