Unanswered Questions
49,207 questions with no upvoted or accepted answers
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Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
- 24.9k
17
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892
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An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
17
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602
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Trying to reconcile two facts about the Appell-Lerch sum learned from Polishchuk and Zwegers
One of the key characters in the thesis of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau)=\frac{e^{\pi i u}}{\vartheta(v;\tau)...
- 18.4k
17
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751
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Elements of finite fields with many powers of trace zero
Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\...
- 767
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427
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On manifolds which do not admit (smooth) actions of finite groups
Question: Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no continuous effective group actions of ...
- 517
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975
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What to do with results you found but cannot prove(outside your research area)?
Not sure if MathOverflow is still a place to discuss such things, but I'll give it a try. Tell me an alternative site, in case it is wrong here. I translated a representation-theory/combinatorial ...
- 26.5k
17
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536
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Question about combinatorics on words
Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...
user6976
17
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1k
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Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?
The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
- 7,817
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737
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Is there an approach to Gabber's theorem from the singular support of coherent sheaves?
David BZ told me that the old theory of singular support of $D$-modules fits into the new theory of singular support of coherent sheaves, via the derived loop space. I wonder how to reconcile that ...
- 4,949
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Does Mochizuki's proof of abc conjecture gives an upper bound for the quality of a triple?
The quality of a triple $(a,b,c)$ of coprime positive integers with $a + b = c$ is defined as
$$q(a,b,c) := \frac{\log(c)}{\log(\mathrm{rad}(abc))}.$$
Then
$$a+b = c = \mathrm{rad}(abc)^{q(a,b,c)}.$$...
17
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224
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GPS calculations under $L^p$ norms
GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...
- 151k
17
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433
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Need explicit formula for certain "$q$-numbers" involving gcd's
The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
- 18.4k
17
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604
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Bunnity of multilinear maps
Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
- 12.4k
17
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808
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Decidability of $x^3+y^3+z^3 = c$
I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^...
- 171
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674
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The topos for forcing in computability theory
My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site."
My ...
- 6,287
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255
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Approximation of the effective resistance on Cayley graph
Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
- 395
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488
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Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
17
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1k
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What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?
For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...
- 149k
17
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824
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What's the big deal about $M_{13}$?
$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
- 11.2k
17
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Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k
17
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553
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Lie algebras vs. graph complexes
A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
- 2,035
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420
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Do the coefficients of these irreducible polynomials always become periodic?
Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...
- 13.4k
17
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953
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A functor of points approach to Berkovich analytic spaces
Is it possible to define a Berkovich analytic space via its functor of points?
Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...
- 171
17
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757
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The spectral sequence of a tower of principal fibrations
Assume we have a tower of fibrations (of simplicial sets, let's say):
$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$
Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
- 15.2k
17
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513
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Maximum automorphism group for a 3-connected cubic graph
The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
- 37.7k
17
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449
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Splay trees and Thompson's group $F$
( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...
- 992
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612
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The Grothendieck Ring of Higher Stacks
The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + [Y]...
- 1,273
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896
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Is there a n/2 version of the Erdős-Hanani conjecture?
This question comes out of REU research from this past summer. Unfortunately weeks of thought led to only trivial observations and the conclusion that the problem is quite hard.
Fix $k,t$. Let $F$ be ...
- 24.2k
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731
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Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
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969
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Groups generated by 3 involutions
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...
- 19.6k
17
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585
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Actions on ℍⁿ generated by torsion elements
Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...
- 45k
17
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answers
782
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Lubin-Tate vs cohomological approach to local CFT
Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...
- 171
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What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
- 47.3k
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547
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Does a symplectic group act on a tensor power of a spin representation?
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$.
Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
- 9,132
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788
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How many Hecke operators span the Hecke algebra?
This is a generalisation of my earlier question about generators for the level 1 Hecke algebra.
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. ...
- 37k
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Special values of Artin L-functions
This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...
- 13k
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Combinatorial identity involving the Coxeter numbers of root systems
The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
$\...
- 2,449
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monomorphisms and epimorphisms of local rings
I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...
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Vector Bundles on the Moduli Stack of Elliptic Curves
As is well known, there is classification of line bundles on the moduli stack of elliptic curves over a nearly arbitrary base scheme in the paper The Picard group of $M_{1,1}$ by Fulton and Olsson: ...
- 12.1k
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What is known about module categories over general monoidal categories?
All of the literature I have seen on module categories over monoidal categories has been in the rigid $k$-linear semisimple case, more or less in the spirit of Ostrik's paper,
Ostrik, V. Module ...
- 7,237
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1
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How are the two natural ways to define “the category of models of a first-order theory $T$” related?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Elem{Elem}$Background/Motivation: Inspired by an interesting question by Joel, I’ve been wondering about the relationship between two very natural ...
- 3,966
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310
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Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
- 3,577
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218
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If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
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188
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Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
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519
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Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 889
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0
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426
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Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
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434
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University library dropping independent journal subscriptions. What to do?
We received an email from our university library stating that they plan to drop subscriptions to the following journals. These are some of the best journals published by independent sources. Of course,...
- 2,751
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429
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Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes
I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes".
The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5.
I ...
- 161
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772
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Ideas behind Gitik's solution of PCF conjecture
Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
- 32.2k
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Why did Bourbaki not use universal algebra?
I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra?
Well, universal algebra is not much older than category ...