**14**

votes

**2**answers

977 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.
...

**9**

votes

**1**answer

485 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 ...

**36**

votes

**0**answers

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 $l^2$...

**8**

votes

**1**answer

576 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 ...

**19**

votes

**2**answers

3k 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 ...

**10**

votes

**1**answer

533 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 ...

**2**

votes

**0**answers

326 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, ...

**10**

votes

**1**answer

327 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

**2**answers

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

**2**answers

346 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 $...

**29**

votes

**5**answers

1k 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
...

**17**

votes

**10**answers

7k 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 ...

**3**

votes

**1**answer

659 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 ...

**14**

votes

**1**answer

582 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 ...

**9**

votes

**4**answers

1k views

### problems from the scottish book

Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...

**6**

votes

**1**answer

304 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 $f(...

**1**

vote

**1**answer

713 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 $...

**6**

votes

**0**answers

401 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 ...

**13**

votes

**1**answer

606 views

### Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...

**11**

votes

**7**answers

2k views

### Open problems in the theory of compact quantum groups

What are the important open problems in the theory of compact quantum groups? Or conjectures?
Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...

**10**

votes

**4**answers

2k views

### What are the major open problems in design theory nowaday?

I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?

**20**

votes

**2**answers

1k views

### Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...

**22**

votes

**4**answers

2k views

### Open problems in Birational Geometry, after BCHM

Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular ...

**8**

votes

**1**answer

806 views

### Transcendentality of all irrationals in the Cantor set

Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...

**46**

votes

**1**answer

2k views

### A function whose fixed points are the primes

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...

**14**

votes

**2**answers

839 views

### Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$...

**2**

votes

**2**answers

346 views

### If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ?
...

**7**

votes

**1**answer

639 views

### approximate uncertainty principle for finite abelian groups

Edit: we cannot find such an example. It would imply a negative solution to the KS${}_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in this paper.
In fact, their ...

**12**

votes

**1**answer

488 views

### A maximal element, where Schur gives a minimal element

Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices. If $H$ is a subgroup of $\frak ...

**7**

votes

**0**answers

252 views

### Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
ﬁnite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...

**11**

votes

**3**answers

482 views

### Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$.
In which cases the conjecture is known ...

**11**

votes

**1**answer

488 views

### Are Galois groups of Q with restricted ramification supposed to be finitely generated?

Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether ...

**19**

votes

**0**answers

459 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'...

**7**

votes

**0**answers

229 views

### Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
...

**5**

votes

**2**answers

426 views

### Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda_{\beta,\...

**8**

votes

**3**answers

944 views

### Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...

**10**

votes

**1**answer

814 views

### The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...

**3**

votes

**2**answers

439 views

### S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.

**9**

votes

**1**answer

543 views

### Minimizing the excursion of a sum of unit vectors

I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...

**1**

vote

**0**answers

230 views

### Sums of Strongly z-ideals

In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...

**1**

vote

**1**answer

531 views

### A question concerns prime numbers

I have arrived to this conjecture in my work, I am not sure that is true or false. So I would appreciate if someone give a counterexample or prove it.
My question: Let n be a non-prime such that n-1 ...

**2**

votes

**0**answers

322 views

### weak hopf conjecture

Hi:
I am thinking of the following problem which is related to weak Hopf conjecture:
Let $E$ be the total space of a vector bundle over a compact nonnegatively curved manifold $B$. Let k=the rank ...

**16**

votes

**0**answers

438 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, ...

**7**

votes

**1**answer

252 views

### Sparse ramsey theory

It is known that for any graph H and all $k∈N$, there exists a graph $G$ such that any $k$-coloring of the edges of $G$ yields a monochromatic copy of H and ω(G)=ω(H) (the two graphs have the same ...

**12**

votes

**10**answers

3k views

### Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...

**45**

votes

**30**answers

6k views

### Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...

**8**

votes

**1**answer

284 views

### Maximal class of simple graphs of order $n$ with mutually distinct numbers of spanning trees

This problem in some ways related to this post.
Let $A_n$ be the set of all integers $x$ such that there exist a connected simple graph of order $n$ having precisely $x$ spanning trees. Study the ...

**25**

votes

**1**answer

1k views

### Important open questions in the field of Tropical geometry

What are some of the important unanswered questions in the field of tropical geometry?

**43**

votes

**2**answers

1k views

### Local structure of rational varieties

I've been asked this question by a colleague who's not an algebraic geometer; we both feel that the answer should be "no", but I don't have a clue how to prove it.
Here's the question:
let $X$ be a ...

**17**

votes

**3**answers

3k views

### How to Tackle the Smooth Poincare Conjecture

The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. ...