Mathematical methods for the many-body problem in statistical and quantum mechanics – MaThoStaQ

The goal of this project is to elaborate mathematical methods applicable to various physical situations in which a large number of interacting particles are involved. The main difficulty in the study of the many-body problem is the possibility that particles be correlated in a complicated manner in order to reduce their interaction energy. A major trend in many-body physics is thus to suppose a particular form to the admissible correlations, which leads to effective theories of which the mean-field approximations, where one assumes particles to be independent, are the simplest. Important and fundamental questions consist in quantifying the validity of these approximations, looking for corrections to a mean-field description, and imagining different approaches when the system at hand is intrinsically strongly correlated. We will focus on static and thermal properties of large systems of classical or quantum particles, working in parallel on three themes: "Crystallization problems in classical Coulomb systems", "Effective theories for many-body bosonic systems", "Strongly correlated quantum Hall phases". Interactions between particles play a crucial role in the situations we want to study, and our aim is to get some control of them by means of rigorous mathematical methods. Our ambition is to obtain new methods and results of both physical relevance and mathematical interest.