Recent years have seen important progress in the understanding of two dimensional statistical physics. The rigorous study of macroscopic random geometric structures generated by microscopic random inputs and interactions has been greatly stimulated by the introduction of Schramm-Loewner Evolutions (SLE) and related objects. This enabled to establish results predicted by Conformal Field Theory (CFT) in Theoretical Physics, most often without matching the techniques. The P.I.'s research program is primarily directed at realizing CFT concepts from SLE based constructions. The methods and issues involve an interplay of geometric, functional analytic and representation theoretic aspects.
The goal of this proposal is to analyze mathematical models of the physical phenomenon of phase transition. A phase transition describes a sharp qualitative change in a physical system under variation of an external paramater, such as freezing of water (transition from liquid to solid phase as temperature decreases). At the phase transition, a random macroscopic geometry may emerge, akin to the wiggly interfaces separating non mixing fluids like oil and water. The study of those fluctuating interfaces is rooted in both Probability Theory and Theoretical Physics. Special focus will be placed on the interaction between these two approaches, and other areas of mathematics involved in the analysis.
