9 problems

Let $A$ denote the lattice-area parameter in the Lennard-Jones minimization problem … The triangular lattice, rhombic lattices, a unique minimizer, and rectangular lattices are the…

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…

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 $E_f(L)$ 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…

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…

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

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 $\zeta_L$ 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,…