# 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?"

494
questions

**-3**

votes

**0**answers

180 views

### problems in algebraic geometry

In the section 6 Nonsingular Curves of the "Algebraic Geometry by Robin Hartshorne " I study :
"In considering the problem of classification of algebraic varieties, we can
formulate ...

**3**

votes

**0**answers

196 views

### Dense sets in $\Bbb{R}^2$ with rational distance

We call a subset $S\subset \Bbb{R}^2$ rationally distanced if all $s_1,s_2 \in S$ have rational Euclidean distance.
The Erdos-Ulam conjecture asks if there is a dense subset of $\Bbb{R}^2$ which is ...

**3**

votes

**0**answers

255 views

### On RH in the Clay Institute list

As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...

**0**

votes

**0**answers

87 views

### How small the radical of $xyz(x+y+z)$ can be infinitely?

This is an open problem.
Let $x,y,z$ be coprime integers (not necessarily pairwise coprime)
and no proper subset sum of $\{x,y,z,-(x+y+z)\}$ is zero.
For a quadruple $(x,y,z,-(x+y+z))$ define the ...

**3**

votes

**2**answers

201 views

### Open problems in matroid theory

I read Oxley's book on matroid theory and found the theory fascinating. At the end, Oxley stated some open problems and conjectures in matroid theory.
Are there any modern lists about such problems?
...

**2**

votes

**0**answers

236 views

### Is function $f: \text{Aut}_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])\to\mathbb{Sets}$ with such properties surjective?

$\mathbb{Sets}$ is the set of all finite subsets of $\mathbb{C}$ including the empty set.
Sets $S, S'\in\mathbb{Sets}$ are called equal mod some set $F\in\mathbb{Sets}$ if $S\cup F = S'\cup F$
Let $G =...

**3**

votes

**0**answers

247 views

### How to put a monetary incentive on an open research problem? [closed]

Since the old days, many famous mathematicians have been attaching monetary rewards to problems. I would like to put monetary bounties on some research mathematics problems. Some of those problems ...

**7**

votes

**2**answers

658 views

### Profound but not popular mathematical topics and notions

The algebraic Theory of Invariants used to be a hot topic until David Hilbert proved his two theorems about invariants. Then for tens of years, the popularity of the topic went down a long time before ...

**0**

votes

**0**answers

79 views

### What it is wrong with the proof $PH = AM = coAM$ based on FORMULA ISOMORPHISM?

Confusion is possible, but Emil's comment and a paper imply
major result: collapse of the polynomial hierarchy.
Q1 What is wrong with the proof that $PH = AM = coAM$ given below?
In a comment
about ...

**2**

votes

**0**answers

93 views

### Two more counterexamples to a conjecture from 1975 about hamiltonicity of digraphs

Question from 2013
gives one counterexample to Nash-Williams's conjecture 1975 about hamiltonicity
of dense digraphs.
In the linked answer, @LouisD "reverse engineered" the counterexample
...

**5**

votes

**2**answers

535 views

### Foundational results dependent on/equivalent to the continuum hypothesis or its negation?

I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:
If $\{ X_i \}_{i \in I}$ is any ...

**5**

votes

**0**answers

81 views

### Frechet-Urysohn quotient of second countable locally compact Hausdorff space

In this paper from 2010 https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1001/arhangav.pdf Arhangelskii asks if there is a quotient of a second countable locally compact Hausdorff space which is Frechet-...

**5**

votes

**0**answers

162 views

### Heuristic for a density conjecture related to the Collatz $(3x+1)$-problem

First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as:
$$
T(n) = \left\{ \begin{array}{}
3n+1, & \text{if $n$ is odd}\ \\
n/2, & \text{if $n$ is even}
\end{array} \right.
$$
...

**9**

votes

**2**answers

255 views

### Generalized figures of constant width

Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$?
This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...

**-2**

votes

**2**answers

241 views

### Basic research problems references [closed]

I have been looking for research problems in pure mathematics that I can try to solve for publishing papers. I am quite aware that it takes a lot of time and effort to get to a level where I can do ...

**11**

votes

**2**answers

589 views

### Which curves and surfaces are realizable by linkages? references?

Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 ...

**3**

votes

**0**answers

63 views

### Effective radius of section of a convex set compared to that of the convex itself

The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai
Volumes of Sets of Constant Width in High Dimensions)
to be the radius of the $n$-ball that has ...

**42**

votes

**10**answers

4k views

### List of long open, elementary problems which are computational in nature

I would like to ask a question of a similar vein to this question.
Question: I'm asking for a list of long open problems which are computational in nature which a beginning graduate student can ...

**7**

votes

**0**answers

200 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?

**0**

votes

**1**answer

403 views

### Has Pillai conjecture been proven?

I found the paper https://hal.archives-ouvertes.fr/hal-00698687v9/document which claims the proof of Pillai conjecture.
However, it is not mentioned anywhere that it has been proved. It's stated ...

**5**

votes

**2**answers

588 views

### Conjectures and open problems in representation theory [closed]

Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?

**34**

votes

**1**answer

2k views

### Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?

[I am a co-moderator of the recently started Open Problems in Algebraic Combinatorics blog and as a result starting doing some searching for existing surveys of open problems in algebraic ...

**7**

votes

**0**answers

283 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 ...

**13**

votes

**2**answers

325 views

### Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?

Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...

**12**

votes

**0**answers

294 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 ...

**27**

votes

**7**answers

4k views

### Decision problems for which it is unknown whether they are decidable

In computability theory, what are examples of decision problems of which it is not known whether they are decidable?

**7**

votes

**1**answer

514 views

### A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u )

Edit: According to answer and comments by Prof. Valette we edite the question.
The Kadison Kaplansky conjecture says:
Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\...

**8**

votes

**1**answer

361 views

### What is the state of research on finding all Prime Knots with 17 Crossings?

In this 1998 journal paper, all the prime knots with 16 or fewer crossings are found (some of which were found earlier by others). There are over 1.7 million such knots. But the prime knots with 17 ...

**13**

votes

**1**answer

376 views

### Which knot invariants have no known diagram-independent descriptions?

Many knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves. Now in principle, any knot invariant can be described ...

**133**

votes

**0**answers

7k views

### Why 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 $Q(x)$ ...

**1**

vote

**1**answer

63 views

### The Total Graph is similar to a line graph

Consider the total graph of a regular graph. From the structure, it seems that it has a similar structure to the line graph ( two different sub-cliques joining at a single point) except that, in ...

**2**

votes

**0**answers

215 views

### How many solutions of x^3 +y^3 = z^3+3 are known? [duplicate]

I've been told that there is reason to think that the equation
$x^3 + y^3 = z^3 + 3$ has solutions in positive integers other than
$$4^3 + 4^3 = 5^3 +3.$$
Can someone tell me the current status of ...

**8**

votes

**0**answers

457 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_{|...

**7**

votes

**1**answer

158 views

### Hyperbolic groups and spaces of negative curvature

Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (...

**0**

votes

**0**answers

152 views

### On the convergence of $\sum_{n\geq 1} \frac{\sin (2^n)}{n^s}$

What is the radius of convergence of the aforementioned series ? If I recall correctly, I once saw a post here on MO claiming that it converges for $\Re(s) > 1/2$, but I can't seem to find the post ...

**13**

votes

**0**answers

615 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 ...

**0**

votes

**0**answers

58 views

### Coloring a graph formed by cliques sharing at most one point

Consider a graph formed by $k$ $k$ order cliques sharing at most one point. Consider thedegenerate case of all cliques disjoint, which is trivially $k$ colorable. Now, to colour any other such graph, ...

**0**

votes

**0**answers

54 views

### Is the exact solution of the wave equation for the scattering of waves by a disk/cylinder an open problem?

The solution exact solution of the Helmholtz equation for the scattering of waves by a sphere is relatively straightforward and has been known since the time of Lord Rayleigh.
The exact solution of ...

**9**

votes

**1**answer

612 views

### Open problems concerning all the finite groups

What are the open problems concerning all the finite groups?
The references will be appreciated. Here are two examples:
Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ...

**28**

votes

**2**answers

717 views

### Open problems in Sobolev spaces

What are the open problems in the theory of Sobolev spaces?
I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to ...

**32**

votes

**11**answers

3k views

### Open questions about posets

Partially ordered sets (posets) are important objects in combinatorics (with basic connections to extremal combinatorics and to algebraic combinatorics) and also in other areas of mathematics. They ...

**106**

votes

**34**answers

16k views

### Noteworthy, but not so famous conjectures resolved recent years

Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch ...

**4**

votes

**1**answer

410 views

### Open Problems in Random Graphs [closed]

I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...

**7**

votes

**0**answers

228 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 ...

**2**

votes

**0**answers

136 views

### Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...

**7**

votes

**3**answers

621 views

### For what sets does the Lebesgue Differentiation Theorem hold in one dimension?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...

**15**

votes

**2**answers

735 views

### Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?

Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...

**29**

votes

**1**answer

1k views

### Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...

**4**

votes

**2**answers

217 views

### Integral equality of 1st intrinsic volume of spheroid

Computations suggest that
$$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$
The question ...

**107**

votes

**1**answer

31k views

### What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...