4 problems

Let $\mathcal A$ be a finite alphabet with at least two elements, and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a two-dimensional configuration. For $k,n\in\mathbb N$, let…

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…

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 $\eta\in\mathcal{A}^{\mathbb{Z}^2}$, 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…