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?"

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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 $Q(x)$ ...
Sil's user avatar
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76 votes
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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\...
Ady's user avatar
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47 votes
0 answers
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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 $l^2$...
Nik Weaver's user avatar
37 votes
0 answers
1k views

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 ...
Mohammad Ghomi's user avatar
32 votes
0 answers
2k views

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 $T \in {\cal F}_k$, ...
Gil Kalai's user avatar
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30 votes
1 answer
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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 ...
Andrew Ranicki's user avatar
27 votes
0 answers
2k views

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 $B \subset A$. ...
Andreas Thom's user avatar
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25 votes
0 answers
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 ...
Anton Petrunin's user avatar
24 votes
0 answers
986 views

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 ...
domotorp's user avatar
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24 votes
0 answers
800 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{...
Chandan Singh Dalawat's user avatar
22 votes
0 answers
1k views

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 ...
William DeMeo's user avatar
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 ...
user avatar
19 votes
0 answers
562 views

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'...
HJRW's user avatar
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17 votes
0 answers
580 views

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, ...
Anton Petrunin's user avatar
17 votes
0 answers
1k views

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 ...
Alex B.'s user avatar
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17 votes
0 answers
1k views

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 ...
Bill Johnson's user avatar
16 votes
0 answers
695 views

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 ...
Sridhar Ramesh's user avatar
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})...
Chris Gerig's user avatar
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16 votes
0 answers
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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? ...
alvarezpaiva's user avatar
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16 votes
0 answers
525 views

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 ...
Ben Webster's user avatar
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16 votes
0 answers
2k views

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 ...
David Steinberg's user avatar
16 votes
0 answers
1k views

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 ...
Gil Kalai's user avatar
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15 votes
0 answers
1k views

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 ...
Kaveh's user avatar
  • 5,362
13 votes
0 answers
401 views

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 ...
Keshav Srinivasan's user avatar
13 votes
0 answers
705 views

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 ...
Ali Taghavi's user avatar
13 votes
0 answers
701 views

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 ...
Aleksei Lissitsin's user avatar
13 votes
0 answers
2k views

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 ...
Kaveh's user avatar
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12 votes
0 answers
595 views

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 ...
Maritza Sirvent's user avatar
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 ...
Jorge Vitório Pereira's user avatar
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 ...
Mohammad Ghomi's user avatar
10 votes
0 answers
321 views

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 ...
Joseph O'Rourke's user avatar
10 votes
0 answers
1k views

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 ...
paul Monsky's user avatar
  • 5,412
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," ...
Joseph O'Rourke's user avatar
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, ...
Ben Golub's user avatar
  • 1,058
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?
D. N.'s user avatar
  • 283
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 ...
interregno's user avatar
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 $3n+1$...
Turbo's user avatar
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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_{|...
Ali Taghavi's user avatar
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 ...
user avatar
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 ...
Erfan Khaniki's user avatar
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 < ...
Scott Morrison's user avatar
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,...
S-F's user avatar
  • 155
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?
Markus's user avatar
  • 1,361
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 ...
Keshav Srinivasan's user avatar
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 ...
Piotr Hajlasz's user avatar
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 ...
Idonknow's user avatar
  • 603
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 ...
Ali Taghavi's user avatar
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 ...
Sergiy Kozerenko's user avatar
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 ...
dwan's user avatar
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