Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.We will first review these motivations, then present the ''mean-field'' derivation of effective models and equations describing the system at the macroscopic scale.

January 16, 12:00-13:00, Seminar room 209, Manchester building.

Abstract: I plan to show a proof of the statement  "residually-finite-by-residually-finite extensions are weakly sofic". The proof is based on characterization of weakly sofic groups 
by solvability of equations over groups. It looks different from other proofs of soficity and weak soficity. I plan to discuss relations of equations on and over groups with soficity in some details.

We consider a system of N points evolving according to the gradient flow of their Coulomb or Riesz interaction, or a similar conservative flow. By Riesz interaction, we mean inverse power s of the distance with s between d-2 and d where d denotes the dimension. We show a convergence result as N tends to infinity to the expected limiting evolution equation.  This was previously an open question in general dimension, rendered difficult by the singular nature of the interaction. We will also discuss briefly similar results in the context of models of superfluidity and superconductivity.