In order to explain the emergence of periodic patterns like atoms in solids or vortices in superconductors, mathematicians have been studying simplified models from a variational point of view. Finding energy minimizers and exploring the possible ground states of such systems is known as Mathematical Crystallization. The purpose of this talk is to review the main crystallization results that have been shown since the eighties. From the Heitmann-Radin Theorem to the recent result we obtained with De Luca and Petrache on the optimality of the square lattice for two-body potentials, I will briefly explain the main ingredients of the proofs as well as the important open problems arising in this field. Connections will be made with other related problems like the asymptotic expansion of the logarithmic energy on sphere, the Jellium energy as well as the question of Cohn-Kumar's Universal Optimality. Furthermore, I will also review some old and recent results about lattice energy minimization, i.e. minimization of pair energies among simple lattices, and associated open questions.

Meeting ID: 814 591 7621 Passcode: 147983