Conformal Stochastic Geometry, Dyson Gas, Potential Theory and Conformal Field Theory

University Of Chicago, Chicago IL

Investigators

Abstract

The award supports research and education in the emerging area of stochastic (or random) conformal geometry and its applications to physics and complex analysis. In recent years several subjects in physics and mathematics have received a tremendous boost by focusing on problems of stochastic conformal geometry - statistics of random shapes, loops, paths and surfaces whose probabilistic measure is largely determined by conformal symmetry. These problems are of interest to various fields in physics and mathematics, especially modern complex analysis where respective insights and methods are often shared. The class of problems described in the proposal is probabilistic in nature, and the use of probabilistic methods is essential. On the other hand, conformal stochastic geometry can be seen as "quantum" extension of classical conformal analysis, therefore methods of modern potential theory and complex analysis are yet another essential part. A unifying theme of the proposal is Dyson gas and Dyson diffusion. In physics Dyson gas appears in Quantum Hall Effect, in Calogero models describing particles with fractional charge and statistics, quantum chaos and random matrix theory. In mathematics Dyson gas appears in the theory of Selberg integrals, complex analysis, the theory of orthogonal polynomials and integrable models of statistical mechanics. In recent years it became clear that Dyson gas has an intrinsic relation with conformal stochastic geometry and representations of Virasoro algebra. Since Dyson's gas can be studied by methods of classical analysis one of the aspect of the target of the proposal to use Dyson gas as a plat to form to attack difficult problems of conformal geometry rigorously. The notion of stochastic geometry has originated in the field of critical phenomena in studies of random interfaces starting and ending on boundaries, but recent quest for statistics of random geometrical objects such as critical fluctuating clusters, critical surfaces, random self-avoiding walks and randomly growing patterns showed that methods and concepts of conformal geometry are at the core of a broader class of developing fields of physics and mathematics. Among them unstable fluid flows, random Gaussian fields, quantum gravity, random matrices, non-equilibrium growth processes including the diffusion limited aggregation, critical phenomena in systems with quenched disorder and Fractional Hall effect. Together, these problems constitute a newly emerging field of conformal stochastic geometry. An important feature of the field of conformal stochastic geometry is a natural interaction of mathematical and physical approaches, methods, and intuition. The proposal builds on the momentum of such unification by focusing on interdisciplinary problems of current interest that fall into the realm of conformal stochastic geometry. Some of them have accurate mathematical formulations; others have not been so far formulated in a rigorous way. One of our goals is to develop mathematical formulations of important physical problems in the domain of conformal stochastic geometry. The broader impact of the proposed research lies in bringing together ideas from various fields of physics and mathematics including complex analysis, probability theory, random matrices, conformal field theory, non-equilibrium growth phenomena, and disordered systems. The proposed research will result in bringing these fields closer by communicating the results to various research communities and promoting collaborations between practitioners in diverse areas. The education and outreach component of the proposal will integrate research into teaching of physics from high school to graduate level. The project will provide research and training opportunities for several graduate students.