The investigator plans to work on the mathematical theory of Dyson's gas, in particular on problems related to conformal field theory. The main topics of the proposed research will be the following. (a) Dyson's gas in two dimensions: classical and quantum Laplacian growth, random normal matrices and general one-component plasma ensembles, bifurcations of the plasma droplets and the universality laws corresponding to different types of singularities, the Polyakov-Alvarez type formulas and the asymptotics of the partition function. (b) Conformal field theory: Dyson's gas approximation of conformal fields; conformal field theory of stochastic Loewner evolutions; SLE curves in Dyson's ensembles.
Dyson's gas is a classical model that relates to a great variety of interesting physical phenomena such as Hele-Shaw flows in hydrodynamics and fractional Hall effect in quantum mechanics. In mathematics, Dyson's gas is an important part of random matrix theory. The model has been intensively studied on the physical level but many fundamental problems remain open on the mathematical side. The investigator will focus on several such problems trying to bring together ideas from various areas of mathematics (complex analysis, probability theory, statistical mechanics) and theoretical physics (conformal field theory, non-equilibrium growth phenomena, disordered systems). The educational component of the proposal consists in the development of new graduate courses and in the organization of two long term research programs. The project will provide research and training opportunities for several graduate students and postdocs.
