10 problems

Wigner crystallization conjecture. The minimum of the jellium energy should be achieved by a crystal, called a Wigner crystal.

Let $\mathbb{W}$ denote the renormalized energy of an infinite configuration of points, and consider its restriction to lattice configurations in low dimensions. Crystallization co…

In two dimensions, consider the renormalized energy for the Riesz gases with power-law exponents in the range under consideration. Abrikosov's conjecture. The renormalized energy i…

Let $\overline{\mathcal{A}}_1$ be the class of admissible electric fields with unit background density, and let … be the minimum value of the renormalized energy. The infimum is at…

Let $\mathcal{W}$ be the renormalized energy and let $\mathcal{A}_{\mu_V(x)}$ denote the class of admissible electric fields associated with the local density $\mu_V(x)$. A Bravais…

Let $\mathcal{W}$ be the renormalized energy for Riesz interactions in dimension $d$, with interaction parameter $s$ satisfying $\max(0,d-2)\le s<d$. A lattice-minimizer conjecture…

Let $W$ be the renormalized energy on discrete subsets of $\mathbb R^2$, and interpret asymptotic density one in the usual planar sense. Sandier–Serfaty conjecture. The triangular…

Let $\mathcal{W}$ be the renormalized energy restricted to lattices, and consider the minimization of $\mathcal{W}$ over the relevant lattice class. Low-dimensional lattice-minimiz…

Perfect-lattice crystallization conjecture. The minimizer of the renormalized energy $\mathcal{W}$ is a perfect lattice.

Vorlatt's hexagonal-lattice conjecture. The minimum of $W$ over $\mathcal{A}_m$ is achieved for currents corresponding to a configuration of points in a perfect hexagonal lattice,…