21 problems

Let … be the hexagonal lattice. A lattice is universally optimal if it minimizes every admissible energy among configurations of the same density. Cohn–Kumar's conjecture. The line…

Let $E_f(L)$ denote the lattice energy of a two-dimensional lattice $L$, and consider the potential $f(r)=e^{-\beta\pi r}-e^{-\alpha\pi r}$, where $\alpha$ and $\beta$ are paramete…

A lattice is universally optimal among lattices if it minimizes energy for every completely monotonic potential among lattices in the relevant Euclidean space, after the prescribed…

Let $\Lambda_2$, $\Lambda_8$, and $\Lambda_{24}$ be the hexagonal lattice in $\mathbb{R}^2$, the $E_8$ root lattice in $\mathbb{R}^8$, and the Leech lattice in $\mathbb{R}^{24}$, r…

Hexagonal crystallization conjecture. The minimum over all two-dimensional lattices of unit density always exists and is always achieved at the hexagonal lattice.

Optimality conjecture for triangular and square lattices. There exist $1<p_1<p_2$ and $1<p_3<p_4$ such that, up to rotation, the following hold:

Minimality for $\textnormal{E}_{\textnormal{BPD}}$ in general. For all $N\in\mathbb N\backslash\{0,1\}$, we have

Let $L$ be a two-dimensional lattice of area $A$, and let $E_f(L)$ denote its lattice energy for a potential $f$. A potential is completely monotone if it satisfies…

Let $s\in(-2,\infty)\setminus\\{d\\}$, and let $e(s)$ denote the minimal energy per unit volume of the Riesz gas. For a Bravais lattice $\mathscr{L}$ with unit-volume fundamental c…

For a Riesz interaction in the Coulomb case $s=d-2$, or for the logarithmic interaction $f(r)=-\log r$ in dimension $d=2$, consider electrons embedded in a uniform background of po…

Face-centered cubic minimizer conjecture. The lattice $\mathsf{D}_3$ is the unique minimizer of

Minimality of the rock-salt structure. There exist $delta_0,delta_1>0$, depending only on $d$, such that the global minimizer of $E_{f_1,f_2}$ is of the form

Rock-salt optimality conjecture. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges satisfying some natu…

Let $\mathsf A_2$ denote the triangular lattice in two dimensions, and let $V$ be a pair potential. Consider the corresponding two-body energy and its asymptotic energy per point f…

Let the face-centered cubic and body-centered cubic lattices in ${\mathbb R}^3$ be scaled to the specified density $\rho$, and let the potential be $r \mapsto e^{-\pi r^2}$. Gaussi…

Let $r_0>0$. For a Bravais lattice $L$ in dimension $3$, let $E_{\alpha,r_0}[L]$ denote the Morse lattice energy, let $\mathcal{L}_3$ be the class of Bravais lattices, and let…

Let $p=(m,n,k)$ range over the sites of the simple cubic lattice $\mathbb{Z}^3$, and assign charges of equal absolute magnitude to these sites, subject to a prescribed total charge…

Let $\zeta_L$ and $\theta_L$ denote the Epstein zeta function and theta function of a Bravais lattice $L$, and let $D_3$ and $D_3^*$ denote the FCC and BCC lattices, respectively,…

Let $E_f[L]:=\sum_{p\in L\backslash\{0\}}f(\|p\|^2)$, where $\|\cdot\|$ is the Euclidean norm in $\mathbb R^2$ and $f:\mathbb R_+^*\to\mathbb R$ is completely monotonic, meaning th…

Triangular-lattice uniqueness conjecture. The triangular lattice $\Lambda_{A_0}$ is the unique solution of $(P)$.

Large-area square-lattice conjecture. If $A$ is sufficiently large, the square lattice is the unique solution of $(P_A)$.