Sylvia Serfaty received her PhD in 1999 from Université Paris Sud. She is currently Silver Professor at the Courant Institute of New York University. She works in the field of partial differential equations and mathematical physics. Her work has particularly focused on vortices in the Ginzburg–Landau model of superconductivity and on the statistical mechanics and dynamics of Coulomb-type systems. Prof. Serfaty authored a book on the Ginzburg–Landau theory with Étienne Sandier, Vortices in the Magnetic Ginzburg–Landau Model in 2007, and is one of the editors-in-chief of the scientific journal Probability and Mathematical Physics. In 2004, she was awarded the EMS Prize for her contributions to Ginzburg–Landau theory. Her other distinctions include the Henri Poincaré Prize in 2012, and the Mergier–Bourdeix Prize of the French Academy of Sciences in 2013. She was elected to the American Academy of Arts and Sciences in 2019. This Medallion lecture will be delivered at the 43rd Conference on Stochastic Processes and their Applications (SPA) in Lisbon, Portugal, July 24–28, 2023: https://www.spa2023.org/
Systems of points with Coulomb interactions
The common point between vortices in superconductors or in fluids, eigenvalues of classical random matrices ensembles, optimal optimization points on the sphere, is that they repel each other logarithmically, i.e. according to the two-dimensional Coulomb interaction. On the other hand, Coulomb gases, particularly in two and three dimensions, have been the object of important studies in statistical physics. Simulations indicate that the macroscopic distribution does not depend much on the temperature, while the microscopic patterns depend strongly on it, and seem, in two dimensions, to converge to triangular lattice patterns as the temperature vanishes. These triangular patterns are the same as those observed in superconductors and superfluids, and are named Abrikosov lattices. Some physics papers also observe numerically a finite crystallization temperature. It turns out that these crystallization questions are directly related to questions in number theory, and to recent breakthroughs by Cohn, Kumar, Miller, and Viazovska on the Cohn–Kumar conjecture.
In this talk, I will review the above motivations for studying large Coulomb systems in any dimension, and present the electric formulation of the energy, how it sheds some light on the microscopic behavior described above and connects to number theory questions, and how it also allows to derive the effective dynamics of many interacting Coulomb particles.