All Questions
17,279
questions
9
votes
4
answers
410
views
Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
9
votes
2
answers
502
views
Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...
9
votes
2
answers
510
views
Quotients of schemes by connected groups
Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...
9
votes
1
answer
844
views
Why is the Fast Fourier Transform efficient?
Is there a conceptual way to understand where the Fast Fourier Transform is avoiding redundant computation and thus achieving $O(n\log n)$ instead of $O(n^2)$.
Consider a standard example of the FFT ...
9
votes
1
answer
549
views
Just a little absoluteness might be cheaper?
Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...
9
votes
1
answer
2k
views
Reference request for a proof of Ramanujan's tau conjecture
In the Wikipedia article it states that Ramanujan's tau conjecture was shown to be a consequence of Riemann's hypothesis for varieties over finite fields by the efforts of
Michio Kuga, Mikio Sato, ...
9
votes
2
answers
448
views
Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?
(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...
9
votes
0
answers
294
views
An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
9
votes
1
answer
1k
views
When are certain group C*-algebras exact?
This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic.
Anyway. There has been a lot of attention given to showing that for certain ...
9
votes
1
answer
1k
views
How do $\infty$-categories allow us to do descent on the derived level?
I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the ...
9
votes
1
answer
525
views
Existence of infinite groups that are too reluctant to be topological
With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
9
votes
2
answers
1k
views
Is every open convex subset of a Riemannian manifold necessarily contractible?
Question: Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?
Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in $...
9
votes
1
answer
1k
views
On Fibonacci numbers that are also highly composite
It is not known if there are infinitely many prime Fibonacci numbers. But can one assert that there is no Fibonacci number >2 that is also highly composite (https://en.wikipedia.org/wiki/...
9
votes
2
answers
623
views
Is $\mathbb{Q}$ the orbit of a rational function under iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...
9
votes
3
answers
788
views
Epimorphisms of relations
Let $\bf Rel$ be the category whose objects are sets and whose morphisms are relations.
What is an epimorphism in this category?
I have a sufficient condition, which is: $R$ is epic if the associated ...
9
votes
0
answers
234
views
Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)
(Previously asked at MSE.)
Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
9
votes
1
answer
922
views
Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
9
votes
3
answers
2k
views
Definition of étale (etc) for non-representable morphisms of algebraic stacks?
I've stumbled upon the statement that the morphism $\pi$ from a root stack of the form $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ (i.e. the "generic" version, not the one concentrated along a divisor) to its ...
9
votes
2
answers
686
views
Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".
Can someone explain what are the major ...
9
votes
5
answers
3k
views
Negative values of Riemann zeta function on the critical line.
From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that:
(1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on ...
9
votes
3
answers
538
views
Product of a Finite Number of Matrices Related to Roots of Unity
Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...
8
votes
1
answer
369
views
Characterizations of Jacobson-Morozov parabolics associated to a nilpotent
Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
8
votes
1
answer
580
views
The Hypercomplex Structure of $SU(3)$
(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...
8
votes
1
answer
588
views
Semisimplicity of the category of coherent sheaves?
The category of coherent sheaves on a locally Noetherian scheme is abelian. Are there some geometric conditions on the scheme that imply that the category of coherent sheaves is semisimple?
Edited ...
8
votes
1
answer
204
views
Computational complexity and commuting functions
EDIT: in this question, I was proposing a conjecture, Prop. 1. Fedor Pakhomov showed a counter-example. In this new question I propose a slightly weaker conjecture that holds even for that example and ...
8
votes
1
answer
1k
views
Surjectivity of a map on inverse limits
(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
8
votes
0
answers
497
views
A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)
Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
8
votes
0
answers
229
views
Intuition for the volume form - combinatorial definition?
I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
8
votes
1
answer
576
views
minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]
Let's consider the space $L^2[a,b]$ of functions on the interval and the norm:
$$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$
Now what if we consider only polynomials with integer coefficients: $f(x) \...
8
votes
3
answers
1k
views
Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?
Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...
8
votes
2
answers
1k
views
Tweetable way to see that Willmore energy is Möbius invariant?
Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional
$$\mathcal{W} = \int_M H^2 dA$$
...
8
votes
2
answers
5k
views
What are the possible numbers of regions that 4 planes can divide space?
What are the possible numbers of regions that 4 planes can create?
We know that the minimum number is 5 and the maximum number is 15.
(http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html)
Is it ...
8
votes
3
answers
482
views
Where does $2\sqrt{d-1}$ come from in Ramanujan graphs?
Ramanujan graphs are the best spectral expanders: $\lambda_2 \le 2\sqrt{d-1}$. I'm looking for some intuition for this value $2\sqrt{d-1}$.
Friedman showed that every random $d$-regular graph ...
8
votes
1
answer
719
views
Why relative consistency results by forcing arguments are provable in finitistic metatheory
It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....
8
votes
2
answers
537
views
Non-trivial examples of Stably diffeomorphic 4-manifolds
I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not ...
8
votes
2
answers
444
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a ...
8
votes
0
answers
2k
views
What is the best lower bound for the domination number in regular graphs of girth 5?
The following theorem is a classical result (see [Alon and Spencer, The probabilistic method, 2nd ed., Theorem 1.2.2]):
Theorem: Let $G$ be a graph on $n$ vertices with minimum degree $d$. Then $G$ ...
8
votes
1
answer
454
views
Cartan formula for Steenrod squares on the cochain level
Steenrod originally defined his squares using explicit cochain-level formulas for simplicial mod-2 cochains. To this end, he introduced higher cup products, which control the failure of the usual cup ...
8
votes
3
answers
3k
views
Cone of curves and Mori theorem for algebraic surfaces
In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" ...
8
votes
1
answer
655
views
Extending the tangent bundle of a submanifold
Let $X$ be a complex manifold, and $Y\subset X$ a compact
submanifold. Is it true that the tangent bundle $TY$ may be
extended (as a holomorphic vector bundle) to some
open neighbourhood of $Y$ in $...
8
votes
1
answer
1k
views
From Shortest Paths to Manifold Structure
I'm relatively green in the differential geometry area, so my apologies if what I'm asking is ill-posed and/or not research-level.
I have a situation where I know the shortest path between any two ...
8
votes
1
answer
2k
views
Integer solution to special system of linear equations
This problem appear in my research, but I am unable to solve it.
There should be an easy argument, but I have not yet found it.
Informal version
An integer $k\geq 2$ is fixed.
We are given a matrix (...
8
votes
2
answers
392
views
Approximation of the identity by simple functions
Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$.
Can we concluded $X$ may be written by a ...
8
votes
2
answers
734
views
Knot complement diffeomorphism groups and embedding spaces
I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of ...
8
votes
1
answer
652
views
Interactions (functors) between equivariant sheaves for different groups?
Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity).
To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
8
votes
1
answer
160
views
How many maximal length Bruhat paths from $u$ to $w$ can there be?
I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
8
votes
3
answers
1k
views
Examples of manifolds that do not admit scalar flat metrics
The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:
(A) Every (smooth) function is a scalar curvature.
(B) The manifold is strongly ...
8
votes
3
answers
726
views
Natural statements independent from true $\Pi^0_2$ sentences
I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
8
votes
2
answers
683
views
Adem relations of Steenrod square without modding out the coboundaries
In the paper Products of Cocycles and Extensions of Mappings,
Steenrod introduced the cup-$i$ product and Steenrod square $Sq^k$:
$$
Sq^k(x_n) \equiv x_n \smile_{n-k} x_n,\ \ \ x_n \in C^n(M^d;\...
8
votes
1
answer
418
views
Cheeger Numbers for 3-regular Graphs
A student wanted a challenging Graph Theory programming project and I had
him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, ...