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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9 votes
2 answers
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Open problems in hyperplane/subspace arrangements?

What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and ...
4 votes
1 answer
669 views

Open problems books [closed]

As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary. The only book of this type that I've come across is ...
Theristo's user avatar
  • 109
6 votes
1 answer
801 views

Open problems in Banach spaces, universality

I have gathered a list of universality problems in Banach spaces which have been solved: 1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces. 2....
18 votes
8 answers
3k views

Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
19 votes
2 answers
6k views

*The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as the open problem in General Relativity? I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper, "Die ...
Joseph O'Rourke's user avatar
4 votes
0 answers
675 views

What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times l!$...
Asvin's user avatar
  • 7,648
2 votes
0 answers
481 views

On Descartes / spoof odd perfect numbers

Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\...
Jose Arnaldo Bebita Dris's user avatar
4 votes
0 answers
312 views

Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? [closed]

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). It seems to me that much ...
Charlie Parker's user avatar
2 votes
1 answer
315 views

Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
Jose Arnaldo Bebita Dris's user avatar
10 votes
2 answers
702 views

Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Question: Can any finite lattice be realized as ...
Sebastien Palcoux's user avatar
19 votes
2 answers
2k views

Open problems in Berkovich geometry

I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
Olórin's user avatar
  • 255
22 votes
5 answers
4k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
user avatar
5 votes
1 answer
607 views

Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So ...
Joseph O'Rourke's user avatar
3 votes
1 answer
1k views

Open problems in compressed sensing

What are the main open problems in compressed sensing? I am interested in theoretical as well as in numerical point of view.
Felix Goldberg's user avatar
19 votes
1 answer
1k views

Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in $\mathbb{P}^3(\...
shamovic's user avatar
  • 431
2 votes
0 answers
125 views

Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$. We know that $C_n$ has a giant component whose size is of ...
HHH's user avatar
  • 21
4 votes
0 answers
170 views

Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following: -- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3) -- Covering ...
Sankeerth's user avatar
6 votes
1 answer
251 views

Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds: Find a pair of complex analytic families $\lbrace M_t\rbrace$ and $\...
Chris Gerig's user avatar
  • 17.1k
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
4 votes
3 answers
819 views

Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that If $K$ is a virtual knot whose underlying Gauss ...
N. Owad's user avatar
  • 313
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
  • 17.1k
31 votes
3 answers
2k views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
Mostafa Mirabi's user avatar
3 votes
1 answer
376 views

reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm. My problem is: I have M auctions and in each auction I have N ...
jhon's user avatar
  • 31
1 vote
1 answer
237 views

A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...
user40780's user avatar
  • 867
10 votes
2 answers
804 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
Zahlendreher's user avatar
  • 1,046
10 votes
4 answers
3k views

Is the Manickam-Miklós-Singhi Conjecture solved? [closed]

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...
Jineon Baek's user avatar
10 votes
1 answer
397 views

Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...
Dongyang Chen's user avatar
13 votes
1 answer
606 views

Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map. Not every ghost map ...
user8463524's user avatar
11 votes
1 answer
350 views

A special case of the integer Hodge conjecture

Let $X$ be a projective complex manifold of dimension $n$. Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic? (We may assume, without loss of generality, that $n=3$, because of the ...
Alex Gavrilov's user avatar
4 votes
0 answers
846 views

Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$.                 Is the ...
Joseph O'Rourke's user avatar
0 votes
1 answer
143 views

Rost Correspondence and minimal Pfister-Neighbors

In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following: Conjecture 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence, then the quadratic form (defining $X$) ...
nxir's user avatar
  • 1,409
20 votes
2 answers
986 views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability $1/g(n)...
Joseph O'Rourke's user avatar
1 vote
1 answer
129 views

simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...
user40780's user avatar
  • 867
6 votes
1 answer
778 views

Is there a precise notion of "almost all" such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...
Sylvain JULIEN's user avatar
1 vote
1 answer
612 views

relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color. However, could we convert edge type of Wang Tile ...
user40780's user avatar
  • 867
47 votes
1 answer
3k views

improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
Jeffrey Shallit's user avatar
6 votes
1 answer
394 views

Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem), "Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
Joseph O'Rourke's user avatar
21 votes
3 answers
2k views

What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-...
Qiaochu Yuan's user avatar
3 votes
2 answers
285 views

Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature: The left color and ...
user40780's user avatar
  • 867
59 votes
1 answer
2k views

Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...
Greg Kuperberg's user avatar
9 votes
1 answer
2k views

What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
Sylvain JULIEN's user avatar
5 votes
6 answers
1k views

practical algorithms for np complete problems

Inspired by: Conjecture on NP-completeness of tesselation of Wang Tile up to finite size And the practicality of this topic (solving tessellation on a lattice): coloring in lattice Computational ...
user40780's user avatar
  • 867
18 votes
2 answers
3k views

Determine or estimate the number of maximal triangle-free graphs on $n$ vertices

Among the collections of the open problems of Paul Erdős on the website of Professor Fan Chung, there is one called "number of triangle-free graphs". http://www.math.ucsd.edu/~erdosproblems/erdos/...
user avatar
8 votes
3 answers
1k views

Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such. Titchmarsh explains in the last chapter ...
Kevin Smith's user avatar
  • 2,470
11 votes
3 answers
905 views

An integral related to the Euler gamma function

The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral $$\eta=2\int\limits_0^1 \...
Zurab Silagadze's user avatar
3 votes
1 answer
255 views

Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
user42388's user avatar
  • 143
10 votes
1 answer
931 views

Ramanujan's problem 754 still open?

In addition to the MO question The Ramanujan Problems. , I would like to ask the following. Problem 754 from the list of the Ramanujan's problems ( http://www.imsc.res.in/~rao/ramanujan/...
Zurab Silagadze'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
15 votes
1 answer
726 views

An infinite, amenable, finitely presentable group with no non-trivial finite quotients

My question is a simple one: is there a group with the properties in the title? In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a ...
HJRW's user avatar
  • 24k
37 votes
2 answers
2k views

A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
Gjergji Zaimi's user avatar

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