48 problems

Let $L\subset\mathbb{R}^3$ range over Bravais lattices of covolume $1$. Define $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha\lvert v\rvert^2}$ for $\alpha>0$ and…

Betke-Henk-Wills conjecture. For any $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$,

Ehrhart's volume conjecture. One should have

Cassels–Swinnerton-Dyer's conjecture. The following are equivalent:

Minkowski-type criterion for three-dimensional lattice coverings. The arrangement $K+\mathbb{Z}^3$ is a lattice covering of $\mathbb{R}^3$ if and only if there exist a unimodular t…

Let $d \geq 2$ and let $A \subset \mathbb{R}^d$ be a finite set with affine dimension $\dim(A)=d$. Stanchescu's conjecture. Then … This conjecture proposes that Stanchescu's constr…

Covering product conjecture. One has

Let $\mathcal B$ be a symmetric set in $\bR^n$ of volume $V=2^{\mathcal O(n)}$, let $k=cn$ for a constant $0<c<1$, let $\mu$ be the Haar distribution on lattices, and let $Y$ be Po…

Binary-form area lower-bound conjecture. One has

Three-dimensional flatness conjecture. No hollow convex $3$-body has width larger than $$ . That is,

A two-dimensional (or, more generally, $n$-dimensional) continued fraction is represented by a sail, whose facets have homeomorphic types, adjacency relations, integer volumes and…

Let $L=C\mathbb{Z}^n\subset\mathbb{R}^n$ be a lattice with Gram matrix $G=C^{\mathsf T}C$ and associated flat torus $T_L=\mathbb{R}^n/L$. Normalize to unit volume, so $\det G=1$. I…

Let $G=\mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime, let $\Bbbk$ be a field of characteristic different from $p$, and let $V$ be a representation of $G$. Let $m$ be the number of…

Let $K$ be a convex body in $\mathbb{R}^{n}$, and let $(K-K)^{\circ}$ be the polar body of its difference body. Makai Jr.'s conjecture. … Equality should hold if and only if $K$ is…

Let a finite collection of lattice vectors spanning $\mathbb{R}^d$ be cosimple if it has a linear dependence whose coefficients are all non-zero and have pairwise different absolut…

Let $Z=\sum_{i=1}^n[\mathbf{0},\mathbf{u}_i]\subseteq\mathbb{R}^{n-1}$ be a lattice zonotope of dimension $n-1$ with $n$ generators, whose every $n-1$ generators are linearly indep…

Let $\mathcal{L}_d$ be the space of function-field lattices and define the Minkowski spectrum by … Here $q$ is the parameter appearing in the function-field absolute value. Singlet…

Let $X_d$ be the space of lattices under consideration, let $A$ be the group acting on $X_d$, and let $\mu=\operatorname{CovRad}_N$ be the covering radius associated with…

For $n\geq3$, let $c(n)$ be the optimal constant defined by … and let $\Delta(K)$ denote the critical determinant. Let $H_{1,\ldots,1}$ be the generalized hexagon in…

Let $K\in\mathcal K^n$ be a full-dimensional convex body and let $\Lambda\in\mathcal L^n$. Let $K_s$ be the symmetrization of $K$, and let…

Let $K\in\mathcal K_{os}^n$ be an origin-symmetric convex body and let $\Lambda\in\mathcal L^n$ be a lattice. Write $K^\star$ and $\Lambda^\star$ for the polar body and dual lattic…

Betke–Henk–Wills–Malikiosis conjecture. One has

Fix an integer $d\geq 2$ and a prime power $q$. Let $\left[\mathcal{L}_d\right]$ be the space of homothety classes of lattices, let $\tilde{\mathcal{K}}^d$ be the ambient vector sp…

Let $d\geq 2$, let $G=\operatorname{SL}_d(\mathbb{R})$, let $\Gamma=\operatorname{SL}_d(\mathbb{Z})$, and let $X_d=G/\Gamma$, identified with the space of unimodular lattices in…

Cassels–Swinnerton-Dyer's conjecture. The condition $N(\Lambda)>0$ is equivalent to the existence of a non-degenerate diagonal operator $D$ such that $D\Lambda$ is an algebraic lat…