Questions tagged [open-problems]

If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"

Filter by
Sorted by
Tagged with
7
votes
2answers
687 views

What is known about polyhedra nets that allow overlapping?

It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
24
votes
7answers
4k views

Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset. Is it true that convex hull of $K$ is compact? Comments: Convex hull of $K$ = intersection of all closed convex sets ...
11
votes
3answers
987 views

Motivation for uniform surjectivity of mod l representations associated to elliptic curves

Background Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $G_{\mathbb{Q}}$ be the absolute Galois group $Aut(\overline{\mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup $E\...
25
votes
0answers
1k views

Milnor's cartography problem

Let $\Omega$ be a round disc of radius $\alpha<\frac{\pi}{2}$ on the unit sphere $\mathbb{S}^2$. It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the ...
10
votes
1answer
884 views

Is every matching of the hypercube graph extensible to a Hamiltonian cycle

Given that $Q_d$ is the hypercube graph of dimension $d$ then it is a known fact (not so trivial to prove though) that given a perfect matching $M$ of $Q_d$ ($d\geq 2$) it is possible to find another ...
7
votes
3answers
1k views

Is Murasugi's conjecture still open?

Normalize the Alexander polynomial (in $t$) so that the positive and negative exponents are balanced. For example in the Conway normalization, make the substitution $z = t^{1/2} - t^{-1/2}$. The ...
12
votes
0answers
570 views

Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree in $\mathbb C^{n+1})$. The fiber over $(\lambda:\mu) \in ...
11
votes
3answers
1k views

level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
16
votes
3answers
2k views

The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points. One high dimensional extension ...
21
votes
4answers
2k views

Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that $$\mathop{\rm area} \Sigma\ge ...
6
votes
1answer
644 views

Infinite dimensional Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes. I heard that this statement is not true in infinite dimensions, ...
11
votes
2answers
690 views

Do torsion-free groups give projectionless group ($C^\ast$) algebras?

One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
31
votes
1answer
3k views

Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...
180
votes
8answers
10k views

Two commuting mappings in the disk

Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(...
40
votes
12answers
2k views

Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...
10
votes
2answers
767 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
0
votes
3answers
1k views

Important (interesting) unsolved problems [closed]

I think it would be interesting to have a list of important unsolved problems in mathematics. What are the important (interesting) problems in your field of work? It would be especially nice, to have ...
20
votes
2answers
2k views

In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
25
votes
3answers
2k views

When does the converse to Schur's Lemma hold?

Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End$_{A-mod}(M)$ is a division ring. A common use is when $R$ is the complex numbers $\...
8
votes
3answers
835 views

Hamiltonian $S^1$ actions with isolated fixed points

I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $...
14
votes
2answers
667 views

Is a smooth closed surface in Euclidean 3-space rigid?

Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically. Robert Connelly found an example of a polyhedral surface that can be deformed ...
76
votes
11answers
25k views

Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...
19
votes
5answers
2k views

Point singularity of a Riemannian manifold with bounded curvature

Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...
18
votes
4answers
1k views

Splitting Pythagorean triples

Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would be least surprised ...
7
votes
0answers
313 views

Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?

Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar. There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...
44
votes
5answers
4k views

Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's? It seems very unlikely,...

1 6 7 8 9 10