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?"

Filter by
Sorted by
Tagged with
-3
votes
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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
2answers
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
2answers
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
0answers
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
10answers
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
2answers
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
0answers
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
7answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
11answers
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
34answers
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
1answer
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
0answers
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
0answers
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
3answers
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
2answers
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
1answer
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
2answers
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
1answer
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 ...

1
2 3 4 5
10