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?"
132
questions with no upvoted or accepted answers
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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
76
votes
0
answers
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2, 3, and 4 (a possible fixed point result ?)
The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...
47
votes
0
answers
2k
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Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
37
votes
0
answers
1k
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
32
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0
answers
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A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"
Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$,
...
30
votes
1
answer
2k
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When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
27
votes
0
answers
2k
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Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
25
votes
0
answers
1k
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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 ...
24
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0
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0's in 815915283247897734345611269596115894272000000000
Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end?
Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way ...
24
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0
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800
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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{...
22
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Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?
If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
21
votes
0
answers
667
views
Avoidable words
Let $u(x_1,...,x_n)$ be a word, $k\in \mathbb{N}$. We say that $u$ is $k$-avoidable if there exists an infinite word in $k$ letters $\{a_1,...,a_k\}$ which does not contain values of $u$ (i.e. words ...
19
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0
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562
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The oriented homeomorphism problem for Haken 3-manifolds
Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
17
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0
answers
580
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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, ...
17
votes
0
answers
1k
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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 ...
17
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0
answers
1k
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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 ...
16
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0
answers
695
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Decidable open problems
Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or ...
16
votes
0
answers
440
views
Does $S^4$ have a "symplecto-homeomorphic" structure?
The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms $(\mathbb{R}^4,\omega_\text{std})\to(\mathbb{R}^4,\omega_\text{std})...
16
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0
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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
16
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0
answers
525
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Comparing the Kazhdan-Lusztig and Steinberg pre-orders
Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
16
votes
0
answers
2k
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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 ...
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 ...
15
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answers
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Razborov's response to Almost Natural Proofs
This post is about Natural Proofs barrier in computational complexity.
There are two recent papers related to this. They are:
Amplifying lower bounds by means of self-reducibility by Eric Allender ...
13
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0
answers
401
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Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?
Googology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ...
13
votes
0
answers
705
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
13
votes
0
answers
701
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Second duals of Grothendieck spaces
The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces.
(See, e.g., this note ...
13
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0
answers
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How can an approach to $P$ vs $NP$ based on descriptive complexity avoid being a natural proof in the sense of Raborov-Rudich?
EDIT: This question has been modified to make it a stand-alone question. Feel free to retract your votes for the previous version.
Here are Vinay Deolalikar's paper, and Richard Lipton's first post ...
12
votes
0
answers
595
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Once differentiable, piecewise degree three polynomials on triangulated planar domains
Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...
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 ...
10
votes
0
answers
253
views
Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
10
votes
0
answers
321
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Bang's open question strengthening Tarski's planks problem
Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...
10
votes
0
answers
1k
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Dissecting trapezoids into triangles of equal area
[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
9
votes
0
answers
248
views
Randomly placing nonoverlapping unit cuboids
Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...
9
votes
0
answers
295
views
Convergence in $L^2$ of iterated expectations
Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...
9
votes
0
answers
893
views
Tarski Monster group with prime 5
Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about 5?
8
votes
0
answers
332
views
Has there been any progress on this open problem about co-well-poweredness of accessible categories?
On the relations between accessible categories and large cardinal axioms, one big example is the following:
Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
8
votes
0
answers
498
views
Landau's century-old problems: Anything comparable?
Landau's four problems
are now over a century old (1912), and each still unsolved.
This seems remarkable, even though he was not the originating author all four
(maybe only the 4th?). Still, he ...
8
votes
0
answers
1k
views
Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?
The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
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
360
views
A question related to Conways 99 graph problem
I have observed that the number of triangles $\frac{vk}{6}$ of a strongly regular graph with parameters $(v,k,1,2)$ is given by the coefficient $2(k-1)$ in the molien series of the "4-D extraspecial ...
8
votes
0
answers
135
views
Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I want ...
8
votes
0
answers
341
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 < ...
7
votes
0
answers
149
views
What is the current status of research on the von Neumann's inequality for $n \ge 3$?
Problem
Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$.
Does a constant $C_n \ge 1$ exist, for which it would be true, that:
$$\forall_{p \in \mathbb{C}[x_1, \ldots,...
7
votes
0
answers
241
views
Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
7
votes
0
answers
356
views
When do two knots have isomorphic fundamental bikeis?
A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...
7
votes
0
answers
253
views
An open problem in Sobolev spaces
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator
$$
E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)
\quad
\text{and}
\quad
E:W^{1,q}(\Omega)\to ...
7
votes
0
answers
311
views
Status of two Banach space theory open problems posted by Pełczyński
In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems.
Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is ...
7
votes
0
answers
516
views
Limit cycles as closed geodesics(2)
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
7
votes
0
answers
542
views
Graphs with graphic imbalance sequences
Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
7
votes
0
answers
473
views
How many values a polynomial map misses?
Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...