Crystallization of point configurations to the Square Lattice in 2D.

Abstract: We consider minimum-energy configurations x_1, …, x_N in euclidean space, minimizing the energy given by the sum of V( |x_i-x_j| ) for all pairs of distinct indices i,j, with V an interaction potential.

Recent breakthroughs (Viazovska Ann. Math. 2017, Cohn-Kumar-Miller-Radchenko-Viazovska Ann. Math. 2017) led to a proof in arXiv:1902.05438 that special lattices in dimensions 8 and 24 form “universally optimal configurations”, i.e. optimize our energy at fixed density for a large class of V’s. This was recently extended (P.-Serfaty, Proc. AMS 2020) to include Coulomb interactions. It is conjectured that in 2 dimensions the triangular lattice is also “universally optimal”. If proved, this would give as a corollary the “Abrikosov conjecture” in superconductivity. For now, “crystallization” in 2D to a triangular lattice is only known for localized one-well potentials V (Theil Comm. Math. Phys. 2006).

I will present a recent result with Betermin and De Luca arxiv:1907.06105, in which we add a new proof of crystallization to the list, and prove for the first time that for a class of one-well potentials V, minimum-energy configurations in the plane form a square lattice instead of a triangular one.

 

Date: Aug 04, 2020 at 16:00:00 h
Venue: Modalidad Vía
Speaker: Mircea Petrache
Affiliation: (PUC)
Coordinator: Claudio Muñoz
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Posted on Jul 28, 2020 in Differential Equations, Seminars