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 ...

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14
votes
2answers
562 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 ...
2
votes
1answer
95 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 ...
19
votes
0answers
229 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 ...
4
votes
1answer
424 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 ...
6
votes
5answers
515 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 ...
15
votes
1answer
505 views

Paul Erdős: 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". ...
6
votes
3answers
801 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 ...
3
votes
2answers
203 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 ...
3
votes
1answer
146 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?)
7
votes
1answer
427 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 ( ...
8
votes
0answers
145 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, ...
13
votes
1answer
278 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 ...
24
votes
2answers
841 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 ...
13
votes
0answers
380 views

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? ...
17
votes
2answers
703 views

Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$, and not an integer (so $b > 1$). Q. Does the sequence $$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, ...
6
votes
4answers
1k views

Interactions of number theoretic conjectures and other fields of mathematics

There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
1
vote
1answer
238 views

Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]

This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...
16
votes
1answer
570 views

A Question on 1, 2 ,3 Conjecture

1, 2, 3 conjecture is well-known: If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers ...
7
votes
4answers
438 views

A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that: (1) $M\subseteq N$ (2) $\aleph_{1}^{N}=\aleph_{1}^{M}$ (3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$ Update: According to Peter's useful ...
6
votes
1answer
518 views

A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency". $HOD$ as an inner model of $ZFC$ lies ...
3
votes
0answers
71 views

A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
18
votes
0answers
312 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
9
votes
2answers
378 views

Is the Steiner ratio Gilbert–Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest network interconnecting $P$ must be a tree, which is called a Steiner minimum ...
10
votes
0answers
688 views

Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem, but the ...
3
votes
4answers
626 views

Product of exponents of prime factorization

Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example, $$p(5184) = p(2^6 3^4) = 24 \;,$$ $$p(65536) = p(2^{16}) = 16 \;.$$ Define $P(n)$ as the number of iterations ...
2
votes
0answers
118 views

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$? Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$. (The function ...
1
vote
0answers
189 views

Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]

I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations: $a^2 + b^2 = ...
6
votes
0answers
303 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 ...
2
votes
1answer
209 views

Passing C through a slot

Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through? i.e. which are the necessary and sufficient conditions for a planar compact set C to ...
5
votes
0answers
264 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?
4
votes
0answers
433 views

Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
2
votes
1answer
133 views

Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ...
11
votes
1answer
487 views

Tiling the square with rectangles of small diagonals

For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...
10
votes
1answer
429 views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
8
votes
1answer
401 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
32
votes
0answers
1k views

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 ...
8
votes
1answer
415 views

A question about the Axiom of Choice

Let AC denote the Axiom of Choice. Let PP denote the so-called "Partition Principle" which states that "If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets of S, then S can ...
18
votes
2answers
1k views

Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
9
votes
1answer
492 views

A conjecture by Euler about $8n+3$

Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum $$8n+3=(2k-1)^2+2p,$$ where $k$ is a positive integer, and $p$ is a prime. I want to know whether there has been ...
1
vote
0answers
168 views

Counting factors: is this approach in the literature on multiperfect numbers?

Does the following approach (or something near it) exist in the number theory literature? I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$ and for this question. First, ...
9
votes
0answers
231 views

Are there only finitely many maximal subfactors of a fixed finite index ?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. Question: are there only finitely many maximal subfactors of a fixed finite ...
9
votes
1answer
290 views

Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ...
12
votes
2answers
1k views

Distinctive property of the primes 17 and 19?

Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$. For the obvious heuristic reasons, for large $n$ one would ...
1
vote
2answers
265 views

A conjecture on closed discrete subset

I am struggling with this old problem, which is also posted here: Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of ...
19
votes
4answers
935 views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
15
votes
10answers
2k views

Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
1
vote
1answer
483 views

Collatz conjecture— finite state machine transducer construction, origination?

wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
7
votes
0answers
350 views

Is it true that every f.g. infinite simple group has exponential growth?

Is it true that every finitely generated infinite simple group has exponential (word-)growth? Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of ...
6
votes
1answer
244 views

Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there ...
1
vote
1answer
501 views

An open problem on general topology

There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others. Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a ...