All Questions
17,348
questions
29
votes
2
answers
2k
views
Making $\mathbb{Q}$-cohomology integral
Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...
- 22.2k
29
votes
5
answers
6k
views
What is the difference between a zeta function and an L-function?
I've been learning about Dedekind zeta functions and some basic L-functions in my introductory algebraic number theory class, and I've been wondering why some functions are called L-functions and ...
- 15.1k
29
votes
4
answers
3k
views
Is there an 11-term arithmetic progression of primes beginning with 11?
i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
- 4,124
29
votes
5
answers
5k
views
Partial sums of multiplicative functions
It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
- 28.7k
29
votes
4
answers
6k
views
Why do people say Gödel's sentence is true when it is true in some models but false in others?
I am a beginner, so this question may be naive.
Suppose we have a (sufficiently strong) consistent first order logic system. Gödel's first incompleteness theorem says there exists a Gödel sentence $g$ ...
29
votes
1
answer
3k
views
Zeros of polynomials with real positive coefficients
The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
$L_n=n^{...
- 7,434
29
votes
12
answers
6k
views
When does 'positive' imply 'sum of squares'?
Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...
29
votes
0
answers
3k
views
Why do polytopes pop up in Lagrange inversion?
I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 489
29
votes
0
answers
2k
views
Did Grothendieck overestimate topoi?
I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines:
Из этих тем ...
29
votes
6
answers
5k
views
Can skeleta simplify category theory?
I am not by any means an expert in category theory. Anyway whenever I have studied a concept in category theory I have always had the feeling that most of the subtleties introduced are artificial.
...
- 14.5k
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
29
votes
4
answers
4k
views
Model structure on Simplicial Sets without using topological spaces
The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
- 1,302
29
votes
6
answers
2k
views
Combinatorial Morse functions and random permutations
This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function ...
- 34.1k
29
votes
6
answers
1k
views
Concise model of modern fiat money and its non-conservation
A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...
- 56.2k
29
votes
1
answer
4k
views
What goes wrong for the Sobolev embeddings at $k=n/p$?
For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{...
- 17.2k
29
votes
6
answers
12k
views
Algorithms for calculating R(5,5) and R(6,6)
Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said:
Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...
- 393
29
votes
4
answers
2k
views
Trees in groups of exponential growth
Question: Let $G$ be a finitely generated group with exponential growth.
Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree?
...
- 25.3k
29
votes
3
answers
3k
views
The non-simplicity of $SO(4)$ and $A_4$
It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...
- 3,741
29
votes
2
answers
3k
views
Does Taranovsky's system of ordinal notations make sense?
Dmytro Taranovsky has a Web page on which he claims to define a system of ordinal notations strong enough to provide an ordinal analysis of full second-order arithmetic. I think (perhaps unjustly) ...
- 30.2k
29
votes
2
answers
2k
views
What theorem constructs an initial object for this category? (Formerly "Integrability by abstract nonsense")
Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...
- 12k
29
votes
3
answers
3k
views
Is the theory of categories decidable?
There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...
29
votes
7
answers
4k
views
When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?
Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
- 115k
28
votes
0
answers
564
views
A sequence potentially consisting of only integers
I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences.
Consider the sequence defined by
$$b_n = \frac{(...
- 7,775
28
votes
2
answers
5k
views
Arnold on Newton's anagram
Arnold, in his paper
The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18
wrote the following:
``...Puiseux series, the theory which Newton, hundreds of years before ...
- 88.9k
28
votes
5
answers
3k
views
Is Weil's bound for Kloosterman sums ever attained?
Weil's bound for Kloosterman sums states that for $(a,b)\not=(0,0)$,
$$
|K(a,b;q)|:=\left|\sum_{x\in\mathbb{F}_q^*}\chi(ax+bx^{-1})\right|\leq 2\sqrt{q},
$$
where $\chi$ is a non-trivial additive ...
- 1,008
28
votes
2
answers
2k
views
morphisms representable by algebraic spaces vs morphisms representable by schemes
So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn ...
- 7,161
28
votes
2
answers
4k
views
Universal property of the tangent bundle
If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the ...
- 61.5k
28
votes
3
answers
876
views
Ordering subsets of the cyclic group to give distinct partial sums
Suppose that you are given a set $S$ of $k$ nonzero elements from $\mathbb{Z}_n$. Is it always possible to order the elements of $S$, say $a_1,a_2,\dots,a_k$ in such a way that the partial sums $a_1,...
- 281
28
votes
3
answers
2k
views
How to show modularity of an elliptic curve?
In the days before [W, TW, BCDT], how did people show that specific elliptic curves over $\mathbb{Q}$ were modular? For instance, I was reading through a paper of Buhler, Gross and Zagier from 1985 on ...
- 3,025
28
votes
4
answers
3k
views
When are two symplectic forms "isotopic"?
I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long ...
- 15.2k
28
votes
0
answers
3k
views
What's the intuition between formal smoothness, etaleness. and unramifiedness?
Let $f: X \to Y$ be a morphism of schemes. Then $f$ is called (EGA IV.17) formally smooth if whenever $T$ is an affine $Y$-scheme and $T'$ a closed subscheme of $T$ defined by a nilpotent ideal (it's ...
- 25.3k
28
votes
4
answers
3k
views
Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Let $x>0$ and $n$ be a natural number. Prove that:
$$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
This question is very similar to many contests problems, but ...
- 1,054
28
votes
6
answers
2k
views
Random Alternating Permutations
An alternating permutation of {1, ..., n} is one were π(1) > π(2) < π(3) > π(4) < ... For example: (24153) is an alternating permutation of length 5.
If $E_n$ is the number of alternating ...
- 22.6k
28
votes
6
answers
2k
views
How fast are a ruler and compass?
This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...
- 513
28
votes
2
answers
1k
views
What are applications of commutativity theorems for rings?
Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...
- 49.6k
28
votes
1
answer
2k
views
Why and how did preschemes become schemes?
Originally (e.g., in the first edition of EGA and in Mumford's Red Book), what are now called "schemes" were referred to as "preschemes." The word "scheme" was reserved for what are now called "...
28
votes
7
answers
10k
views
Roadmap to learning about Ricci Flow?
Hello,
I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
- 289
28
votes
0
answers
657
views
Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
- 20.8k
28
votes
2
answers
1k
views
Derivation of certain sums "the hard way"
It is a well-known fact, that one can derive some spectacular identities, e. g.
$\sum^{n-1}_{m=1}\sigma_3(m)\sigma_3(n-m)=\frac {\sigma_7(n)-\sigma_3(n)}{120}$
$\sum^{n-1}_{m=1}\sigma_3(m)\...
- 3,680
28
votes
6
answers
3k
views
Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
- 1,890
28
votes
4
answers
6k
views
Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
- 1,499
28
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,047
28
votes
1
answer
1k
views
Are entire functions “essentially” determined by their maximum modulus function?
(Note: This has been asked on Math SE, but without an answer after almost two years and one offered bounty.)
For an entire function $f$ let $M(r,f)=\max_{|z|=r}|f(z)|$ be its maximum modulus function. ...
- 490
28
votes
8
answers
4k
views
Is there a compact group of countably infinite cardinality?
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...
- 12.6k
28
votes
3
answers
2k
views
Being a subgroup: proof by character theory
Let me first cite a theorem due to Frobenius:
Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then
$...
- 51.6k
28
votes
2
answers
7k
views
How to solve a quadratic equation in characteristic 2 ?
What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.
...
- 12.1k
28
votes
1
answer
2k
views
Analogy between the exterior power and the power set
The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...
- 61.5k
28
votes
2
answers
2k
views
Is $\mathbb{C}^2$ homeomorphic to $\mathbb{C}^2 - (0,0)$ with the Zariski topology?
A fellow grad student asked me this, I have been playing for a while but have not come up with anything. Note that $\mathbb{C}$ is homeomorphic to $\mathbb{C} - \{0\}$ in the Zariski topology - just ...
- 12k
28
votes
2
answers
5k
views
Which p-adic numbers are also algebraic?
What is $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ ?
For instance, we know that $\mathbb{Q}_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}_p \cap \...
- 1,951