**8**

votes

**1**answer

1k views

### Smallest dilation of a quadrilateral?

What is the smallest dilation of a quadrilateral in $\mathbb{R}^d$?
This may be an open problem, which I know is verboten on MO.
So my question is: Is this indeed open?
It will take me some time to ...

**7**

votes

**5**answers

2k views

### List of recently solved mathematical problems

I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only ...

**55**

votes

**0**answers

3k views

### Volumes of Sets of Constant Width in High Dimensions

Background
The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...

**6**

votes

**6**answers

599 views

### Statements reliant on conjectures

There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.
What other conjectures have a large number of proven consequences ?

**36**

votes

**3**answers

5k views

### Do there exist chess positions that require exponentially many moves to reach?

By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an ...

**8**

votes

**1**answer

469 views

### Periodic Automorphism Towers

In Scott's classic textbook on Group Theory, he asks:
Suppose that $G$ is a finite group. Is the sequence of isomorphism types
of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic?
...

**42**

votes

**2**answers

3k views

### Polynomials having a common root with their derivatives

Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be ...

**49**

votes

**6**answers

9k views

### Is Thompson's Group F amenable?

Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ...

**17**

votes

**1**answer

1k views

### Coiling Rope in a Box

What is the longest rope length L of radius r that can fit into a box?
The rope is a smooth curve with a tubular
neighborhood of radius r, such that the rope does not
self-penetrate. For an open ...

**27**

votes

**6**answers

2k views

### Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a?

For example for n=2 coloring odd numbers red, numbers of the form 4k+2 blue and so on works.
This problem was posed in the KoMaL for n+1 prime, if I know well by Geza Kos. I verified it for all ...

**9**

votes

**0**answers

685 views

### Dissecting trapezoids into triangles of equal area

[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...

**17**

votes

**2**answers

696 views

### Ramsey multiplicity

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ ...

**1**

vote

**0**answers

100 views

### Square powers of hemicontinuous operators

Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line
segment of H to the weak ...

**16**

votes

**3**answers

1k views

### Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...

**6**

votes

**1**answer

686 views

### Current status of Bloch Constant and Landau Constant bounds

The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...

**6**

votes

**1**answer

855 views

### Is there an uncountable, non-discrete, Hausdorff Toronto space?

We call a topological space $X$ a Toronto space if for any subspace $Y \subseteq X$ such that $Y$ and $X$ have the same cardinality it follows that $Y$ is homeomorphic to $X$.
Does anybody know what ...

**7**

votes

**2**answers

727 views

### Covers of Riemann surfaces which become arbitrary close in Teichmuller space

Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g_i \to \infty$ and covers $S_i, S_{i}'$ of $S,S'$, both of genus $g_i$, such that ...

**20**

votes

**4**answers

8k views

### Does pi contain 1000 consecutive zeroes (in base 10)?

The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at ...

**261**

votes

**4**answers

21k views

### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\ $ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?

**57**

votes

**2**answers

3k views

### Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...

**6**

votes

**1**answer

453 views

### distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field

This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...

**45**

votes

**0**answers

2k views

### 2, 3, and 4 (a possible fixed point result ?)

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that $\Vert ...

**67**

votes

**4**answers

4k views

### If 2^x and 3^x are integers, must x be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
If $n^x$ is an integer for ...

**52**

votes

**3**answers

2k views

### Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...

**13**

votes

**1**answer

1k views

### Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?

Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of ...

**3**

votes

**2**answers

821 views

### What's known about the 3rd coefficient in the BMV conjecture?

The most appealing statement of the Bessis-Moussa-Villani conjecture is as follows:
Conjecture: For all Hermitian positive semidefinite $n\times n$ matrices $A$ and $B$,
and all positive integer ...

**7**

votes

**1**answer

2k views

### What's the current state of Yang Mills Mass Gap question?

What's the current state of Yang Mills Mass Gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction ...

**29**

votes

**4**answers

2k views

### Factorials in Pascals Triangle

Hi,
I asked this question of Keith Conrad, and he suggested that I try posting here. One of my students observed that the only instances of factorials in the interior of Pascal's triangle are ...

**15**

votes

**2**answers

1k views

### Are there Ricci-flat riemannian manifolds with generic holonomy?

This may well be an open problem, I'm not sure.
In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete ...

**13**

votes

**1**answer

707 views

### $SL_2 R$ Casson invariant?

Casson's invariant is an invariant of a homology 3-sphere, obtained by
``counting" representations of the fundamental group into $SU(2)$.
I was wondering if there is an analogous invariant counting ...

**9**

votes

**5**answers

2k views

### On Polynomials dividing Exponentials

EDIT: it turns out that no answer to this is known, as the authors of the book it is in have now confirmed they do not know how to do it. Will Jagy.
ORIGINAL: I have been wondering if there exist ...

**10**

votes

**2**answers

1k views

### A generalization of Cauchy's mean value theorem.

The following simple theorem is known as Cauchy's mean value theorem. Let $\gamma$ be an immersion of the segment $[0,1]$ into the plane such that $\gamma(0) \ne \gamma(1)$. Then there exists a point ...

**11**

votes

**0**answers

903 views

### MNOP conjecture

Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...

**23**

votes

**2**answers

758 views

### Is there an associative metric on the non-negative reals?

Recall that a function $f\colon X\times X\to \mathbb{R}\_{\ge 0}$ is a metric if it satisfies
definiteness: $f(x,y) = 0$ iff $x=y$,
symmetry: $f(x,y)=f(y,x)$, and
the triangle inequality: $f(x,y) ...

**2**

votes

**3**answers

411 views

### Distribution of the sum of the $m$ smallest values in a sample of size $n$

Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly ...

**25**

votes

**3**answers

2k views

### Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...

**12**

votes

**2**answers

786 views

### Prime divisors of numbers 2^n + 3

I'm interested in the following problem: do there exist infinitely many prime numbers $p$ such that $p^2|2^{n}+3$ for some natural number $n$?
Some motivation:
If we replace the function $2^n + 3$ ...

**6**

votes

**0**answers

556 views

### Hilbert subspaces of indefinite inner product spaces

Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...

**13**

votes

**0**answers

1k views

### Finite Rank Commutators

My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...

**18**

votes

**2**answers

689 views

### Deligne-Simpson problem in the symmetric group

Question.
Let $C_1,\dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly,
each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles
have the ...

**22**

votes

**3**answers

2k views

### Topologically distinct Calabi-Yau threefolds

In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...

**8**

votes

**2**answers

861 views

### Borsuk pairs of Banach spaces

Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ...

**26**

votes

**4**answers

1k views

### Are most cubic plane curves over the rationals elliptic?

%This is a new version of the original question modified in the light of the answers and comments.
The word 'most' in the title is ambiguous. The following is one way of making it precise.
...

**16**

votes

**0**answers

549 views

### Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.
Question. Is there a smooth proper scheme ...

**14**

votes

**1**answer

1k views

### А generalization of Gromov's theorem on polynomial growth

I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao).
Assume for a group $G$ there is a
polynomial $P$ such that given
$n\in\mathbb N$ there is set of
...

**15**

votes

**0**answers

730 views

### Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality
Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X.
For a real number p between zero and one, we consider a ...

**6**

votes

**0**answers

267 views

### Enumerating (generalized) de Bruijn tori

Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets ...

**3**

votes

**0**answers

555 views

### Artin Schreier Theorem for Rings

This has been in my mind for quite some time. Looking at Artin Schreier Theorem for fields:
If L is a field and K its algebraic closure and if 1< [K:L] < infinity then L=K[i] and L is a real ...

**11**

votes

**6**answers

3k views

### Solving NP problems in (usually) Polynomial time?

Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.
The best example of this is probably the traveling salesman problem, for which extraordinarily large instances ...

**9**

votes

**1**answer

851 views

### Topological “Interpolation” ?

Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...