13 problems

Let $\Lambda$ range over periodic configurations of arbitrary fixed density in phase space, and consider the associated quantum packing, covering, and paving problems. The hexagona…

Consider optimal quantization in three-dimensional Euclidean space, and let $V$ denote the polytope whose scaled copies asymptotically describe the interior Voronoi cells in an opt…

Universal Gaussian Gabor asymptotic conjecture. There exists a constant $C_\Lambda$, depending on the geometry of $\Lambda(1)$, such that

Let a placement of size $p$ be a finite subset of $\mathds{Z}^n$. For placements $X$ and $Y$, define their centroid by … Define their displacement by … The speed of a trajectory wi…

Let a configuration be a translation-equivalence class of finite placements of pieces on the lattice $\mathds{Z}^n$, and define its speed as the maximum displacement per move among…

Let $z\in\mathbb{H}$ describe the lattice shape, let $(a,b)$ describe the relative position of the two component lattices, and let … where … and $\alpha\in[-1,1]$ measures the inte…

Among flat tori of area $1$, let $A_\Lambda(1)$ denote the quantity measuring the lowest temperature, and let $\Lambda_h$ be the hexagonal lattice. Hexagonal-torus conjecture. The…

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the standard Gaussian, let $\delta>1$, and define … For each separable lattice in this class, let $A$ and $B$ denote the lower and upper frame b…

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the standard Gaussian, let $\delta>1$, and define … For each $\Lambda$ in this class, let $A$ and $B$ denote the lower and upper frame bounds of…

Let $\Lambda_1=\frac{2}{\sqrt{3}}\mathsf{A}_2$ be the triangular lattice in dimension $2$, and let $\Lambda$ have the shape of $\mathsf{D}_3$, $\mathsf{D}_4$, $\mathsf{E}_8$, or…

Let $d\in\{2,3,4,8,24\}$ and let $\Lambda$ be respectively $\Lambda_1$, $\mathsf{D}_3$, $\mathsf{D}_4$, $\mathsf{E}_8$, or $\Lambda_{24}$. For $x_2>x_1>d$, let $E_f$ be the Lennard…

Let $f_\varepsilon$ be the potential defined in the source, let $\Lambda_1$ be the triangular lattice, and consider periodic configurations of fixed density. Local minimality conje…

Let $g$ be a Gaussian window and let $\Lambda$ range over lattices with fixed redundancy. Denote by $A$ and $B$ the lower and upper frame bounds of the associated Gabor frame. Simu…