5 problems

Macroscopic-scaling conjecture. In this case, there should exist $s>d$, a nontrivial measure $\mu$ on $\mathbb{R}^d$, and a sequence of minimizers $(X_N)_{N\geq 2}$ of $E_N$ such t…

Smooth-density conjecture. In this regime, a sequence of minimizers $(X_N)_{N\geq 2}$ should converge to a measure $u$ that has a smooth density with respect to Lebesgue measure.

Triangular-lattice universal optimality conjecture. The usual triangular lattice should minimize the interaction energy among infinite configurations of points with density $1$ for…

Let $p^*(q)$ be the threshold from the theorem such that, for $d=1$, $q\geq 2$, $1\leq\lambda<\infty$, and $p>p^*(q)$, the measure $\rho^*=\frac{1}{2}(\delta_0+\delta_1)$ is the un…

Let $W(x)=|x|^{-12}-|x|^{-6}$ be the Lennard-Jones potential, and consider minimising configurations of $N$ particles for the associated discrete interaction energy. Crystallisatio…