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 …

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 $n^x$ …

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 conta …

Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and the area of their union is at most four. Remarks. If the squares …

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 …

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 …

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-symmetri …

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 ine …

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 t …

I was always fascinated by this nice and easy to state question. I wonder if any progress has been made on it (except for proving it for six). Do you know any related results? For …

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 s …

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 cle …

Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a sub …

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximatively equal to the area of the disc. Does the complement of S necess …

Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and differ by on …