51 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…

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…

Call a finite set $S \subset \mathbb{R}^2$ in general position if no three of its points are collinear, and say that $n$ points of $S$ are in convex position if they are precisely…

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 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…

Prime divisibility embedding conjecture. There is a Euclidean motion $T$ such that

Divisibility embedding conjecture. There is a set $N\subset\mathbb{Z}^n$ such that $k\cdot N$ is congruent to $M$.

Embedding conjecture. One can find a Euclidean motion $T$ such that

Integer-coordinate conjecture. Every Heron simplex can be represented with integer coordinates.

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…

Let $n$ be an integer with $n>3$, and consider the proposition immediately preceding this statement, which asserts that the sum of the areas of the Frégier circles associated with…

Let a satisfying cycle determine its dual object in Euclidean space by interpreting its associated tuple and its bitwise rotations as vertices in $\mathbb{R}^n$. A vertex is said t…

Pedal-polygon equivalence conjecture. For integer $n\ge 5$, if $G(n)$ is the set of $n$-gons, then

For four distinct points $x_1,x_2,x_3,x_4$ in $\mathbb{R}^3$, let the expressions denoted by … in the source be the corresponding angular quantities. Four-point comparison inequali…

For four distinct points $x_1,x_2,x_3,x_4$ in $\mathbb{R}^3$, let the inequality denoted by … holds for every such four-point configuration. The paper notes that this inequality ex…

Let $x_1,\dots,x_n$ be distinct points in $\mathbb{R}^3$, and let $D(x_1,\dots,x_n)$ be the Atiyah determinant; for each $i$, write $D(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)$ for the det…

Let $x_1,\dots,x_n$ be distinct points in $\mathbb{R}^3$, and let $D(x_1,\dots,x_n)$ denote the Atiyah determinant constructed from them. Atiyah's non-vanishing conjecture. … The con…

Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$. Suppose we are given $d$ points $x_1, \ldots, x_d\in S^{d-1}$ and a continuous function $f: S^{d-1}\to \mathbb R$. Knaster's conjec…

Let $S^{2}=\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|=1\}$ be the unit sphere, and for an integer $Nge 2$ let $\omega=(\mathbf{r}{1},\dots,\mathbf{r}{N})$ denote an $N$-tuple of distin…

Let $P \subset \mathbb{R}^3$ be a convex polytope of dimension $3$, with vertex set $V$, edge set $E$, and set of (planar, convex) faces $\mathcal{F}$; let $G = (V, E)$ be its edge…

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…

Call a set $C\subset\mathbb{R}^{2}$ a cover if for every rectifiable plane curve $\gamma$ of length $1$ there is a rigid motion $g$ of the plane (a composition of a rotation and a…

Let $n\ge 1$. A convex body in $\mathbb{R}^n$ is a compact convex set with non-empty interior, and $|\cdot|$ denotes Lebesgue measure on $\mathbb{R}^n$. For a convex body…

For a nonempty set $S \subset \mathbb{R}^2$ let … where $|\cdot|$ denotes the Euclidean norm. Say that a set $C \subset \mathbb{R}^2$ covers $S$ if there is an isometry $\varphi$ o…

Let $n\geqslant 1$ be an integer and let $\mathbf{S}^{n-1}=\{e\in\mathbb{R}^{n}:|e|=1\}$ denote the unit sphere in $\mathbb{R}^{n}$. Call a set $E\subseteq\mathbb{R}^{n}$ a Besicov…