1,022 problems

For every integer $n\ge 1$, energy $E>0$, and noise standard deviation $\sigma>0$, let $x_0,\ldots,x_n\in\mathbb{R}^n$ satisfy $\lVert x_i\rVert^2=E$ for all $i$, and let the signa…

Let $M$ be a complete surface and let $f\colon M\to M$ be a $C^{1}$ diffeomorphism satisfying Condition A: $f$ has dense periodic points, there is a $Df$-invariant splitting…

Let $\mathcal{P}_3$ be the class of convex three-dimensional parallelohedra, namely convex polyhedra that tile $\mathbb{R}^3$ by translations. For every $V>0$ and every…

Let $G$ be a connected real semisimple Lie group of real rank at least $2$, let $P<G$ be a minimal parabolic subgroup, and let $\Gamma<G$ be a discrete subgroup. If the action…

For every non-hyperbolic Riemann surface $X$ and every singular hyperbolic metric $g$ on $X$ in the sense of potential theory, the associated monodromy group…

Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine cir…

For every integer $n\ge 2$ and every admissible marked configuration of $n$ complete pairwise disjoint timelike affine worldlines in Minkowski space, let $F_1,\ldots,F_n$ be the as…

The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem

Let $n \ge 1$ and let $K \subset \mathbb{R}^n$ be a convex body, i.e. a bounded closed convex set with nonempty interior. Call a set of the form $sK + v = \{sx + v : x \in K\}$, wi…

For every tame contact manifold $(Y,\xi)$, every contact form $\alpha$ with $\ker\alpha=\xi$, and every compact Legendrian submanifold $\Lambda\subset Y$, there exist $T>0$ and a s…

Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is…

Let … and let $L=H\cup V$ be the L-shaped corridor formed by these two legs of unit width meeting at a right angle. Call a compact connected set $S\subset\mathbb{R}^2$ a moving sof…

For a finite set $S$ of points in the Euclidean plane, say that two distinct points $p,q\in S$ are mutually visible (with respect to $S$) if no point of $S\setminus\{p,q\}$ lies on…

Let $n\ge 2$ and let $S^{n+1}=\{x\in\mathbb{R}^{n+2}:|x|=1\}$ carry its standard round metric of constant sectional curvature $1$. Let $M^{n}\subset S^{n+1}$ be a smooth, closed (c…

Let $M=\mathbb{S}^{2}\times\mathbb{S}^{2}$, the product of two $2$-spheres, regarded as a closed smooth $4$-manifold. For a Riemannian metric $g$ on $M$ and a point $p\in M$, let…

Let $\alpha\in(0,1)$ and let $S\subset\mathbb{R}^{3}$ be a compact, connected, embedded surface without boundary of class $C^{3,\alpha}$ which is convex, i.e. $S$ bounds a convex b…

Let $n \ge 1$ and let $\mathbb{Z}^n \subset \mathbb{R}^n$ be the standard lattice. For a convex body $K \subset \mathbb{R}^n$ (a compact convex set with nonempty interior) write…

Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with…

Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r)\to \infty$ such that every g…

Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does…

Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least…

Does there exist a sequence of finite sets $A_n\subseteq\mathbb R^2$ such that $|A_n|=n$, every subset of four points of $A_n$ determines at least three distinct distances, and the…

Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\epsilon n$ with no th…

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.$ Let $h(n)$ be the maximal number of un…

Draw $n$ squares inside the unit square with pairwise disjoint interiors, and let $f(n)$ be the maximum possible sum of their side lengths. Is $f(k^2+1)=k$?