7 problems

Let $\mathcal A$ be a finite alphabet and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a configuration. For a non-empty set $\mathcal U\subset\mathbb Z^2$, the restriction…

For $\beta>0$, let … and, for $m\geq 2$, define … A configuration is $L^m_\beta$-universally optimal if it minimizes every admissible periodic energy among $2m$-point configuration…

Szabados's conjecture. If $\eta$ is not fully periodic and $\eta=\eta_1+\cdots+\eta_m$ is a $\mathbb{Z}$-minimal periodic decomposition, then, for each $1\leq i\leq m$, a line…

Universal optimality conjecture. For all $N\in2\mathbb{N}$, $\mathsf{A}_2$, $\mathsf{E}_8$, and the Leech lattice are the unique maximizers of $E_f^\pm$ on $\mathcal{A}_N^1$ in the…

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 $\mathcal{A}$ be a finite alphabet and let $\eta\colon\mathbb{Z}^2\to\mathcal{A}$ be a configuration. Let $\mathcal{S}\subset\mathbb{Z}^2$ be a finite set that is the restricti…

Let $\eta\in\mathcal{A}^{\mathbb{Z}^2}$, let $X_{\eta}:=\overline{\mathcal{O}(\eta)}$ be its orbit closure under the translation action, and let $P_{\eta}(n,k)$ be the number of di…