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?"
555
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0
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MNOP conjecture
Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...
- 2,198
35
votes
2
answers
2k
views
Is there an associative metric on the non-negative reals?
Recall that a function $f\colon X\times X \to \mathbb{R}_{\ge 0}$ is a metric if it satisfies:
definiteness: $f(x,y) = 0$ iff $x=y$,
symmetry: $f(x,y)=f(y,x)$, and
the triangle inequality: $f(x,y) \...
- 4,934
2
votes
3
answers
2k
views
Distribution of the sum of the $m$ smallest values in a sample of size $n$
Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly $P[\min_{i=1\...
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votes
3
answers
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Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
- 18.4k
14
votes
2
answers
1k
views
Prime divisors of numbers 2^n + 3
I'm interested in the following problem: do there exist infinitely many prime numbers $p$ such that $p^2|2^{n}+3$ for some natural number $n$?
Some motivation:
If we replace the function $2^n + 3$ ...
- 191
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votes
0
answers
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views
Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 4,040
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votes
0
answers
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Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.1k
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votes
2
answers
1k
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Deligne-Simpson problem in the symmetric group
Question.
Let $C_1,\dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly,
each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles
have the ...
- 4,460
29
votes
3
answers
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Topologically distinct Calabi-Yau threefolds
In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...
- 23.2k
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votes
2
answers
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Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
- 4,040
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votes
4
answers
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Are most cubic plane curves over the rationals elliptic?
%This is a new version of the original question modified in the light of the answers and comments.
The word 'most' in the title is ambiguous. The following is one way of making it precise.
Question1:...
- 1,620
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votes
0
answers
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views
Smooth proper schemes over Z with points everywhere locally
This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.
Question. Is there a smooth proper scheme $X\to\operatorname{...
- 18.5k
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votes
1
answer
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А generalization of Gromov's theorem on polynomial growth
I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao).
Assume for a group $G$ there is a
polynomial $P$ such that given
$n\in\mathbb N$ there is set of
...
- 43.7k
16
votes
0
answers
1k
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Optimal monotone families for the discrete isoperimetric inequality
Background: the discrete isoperimetric inequality
Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$.
For a real number $p$ between zero and one, we consider ...
- 24.2k
6
votes
0
answers
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Enumerating (generalized) de Bruijn tori
Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets (i.e.,...
- 15.3k
4
votes
0
answers
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Artin Schreier Theorem for Rings
This has been in my mind for quite some time. Looking at Artin Schreier Theorem for fields:
If L is a field and K its algebraic closure and if 1< [K:L] < infinity then L=K[i] and L is a real ...
- 2,175
28
votes
7
answers
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Solving NP problems in (usually) Polynomial time?
Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.
The best example of this is probably the traveling salesman problem, for which extraordinarily large instances ...
- 2,353
9
votes
1
answer
991
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Topological "Interpolation" ?
Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...
- 4,040
10
votes
1
answer
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Linear equation with primes
Is there an integer $n$ with an infinite number of representations of the form
$n=2q-p$, where $p$ and $q$ are both primes?
Given a positive integer $k>1$, I would like to know for which (if any) ...
- 576
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votes
3
answers
1k
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distance regular metric spaces
A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
288
votes
7
answers
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Polynomial representing all nonnegative integers
Lagrange proved that every nonnegative integer is a sum of 4 squares.
Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.
Is there a 2-variable polynomial $f(x,y) \in \...
- 23.6k
39
votes
9
answers
3k
views
The shortest path in first passage percolation
Update (January 17): The problem has now been solved by Daniel Ahlberg and Christopher Hoffman. (Thanks to Matt Kahle for informing us.)
Consider a square planar grid. (The vertices are pair of ...
- 24.2k
41
votes
6
answers
7k
views
Number of valid topologies on a finite set of n elements
I've heard that the problem of counting topologies is hard, but I couldn't really find anything about it on the rest of the internet. Has this problem been solved? If not, is there some feature that ...
2
votes
1
answer
501
views
Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 4,040
17
votes
1
answer
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Topological version of Bogomolov’s question
I'm quoting a question from p. 753 of Gromov's recent paper Singularities, Expanders and Topology of Maps:
Does there exist, for every closed oriented $n$-manifold $X_0$, a closed oriented $n$-...
- 66.8k
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votes
5
answers
1k
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Diameter of universal cover
Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or $\pi_1(...
19
votes
4
answers
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Irreducible polynomials with constrained coefficients
Over at the Cafe, after reading about TWF 285, I asked more-or-less
About how many polynomials with coefficients in $\{\pm 1\}$ and of degree $d$ are irreducible?
and that's what I want to ask ...
- 1,977
42
votes
1
answer
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views
Complex vector bundles that are not holomorphic
Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the ...
- 28.8k
27
votes
6
answers
2k
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When shorter means smaller?
Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it ...
7
votes
2
answers
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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 ...
- 2,585
27
votes
8
answers
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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 ...
- 43.7k
12
votes
3
answers
1k
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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[...
- 10.3k
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votes
0
answers
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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 ...
- 43.7k
11
votes
1
answer
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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 ...
- 3,433
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votes
3
answers
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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 ...
- 1,746
12
votes
0
answers
579
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 ...
- 8,738
11
votes
3
answers
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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 $Q$...
- 611
16
votes
3
answers
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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 ...
22
votes
4
answers
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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 ...
9
votes
1
answer
762
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, ...
- 1,909
14
votes
2
answers
863
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 ...
- 5,335
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votes
2
answers
5k
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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 ...
- 25.5k
185
votes
8
answers
12k
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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(...
- 59.8k
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votes
12
answers
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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 ...
- 17.5k
11
votes
2
answers
909
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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 ...
1
vote
3
answers
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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 ...
21
votes
2
answers
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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 ...
- 1,942
26
votes
3
answers
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 $\...
- 3,083
10
votes
3
answers
1k
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 $...
15
votes
2
answers
851
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 ...