The investigator will study spatial problems of probability theory, with a particular focus on random surfaces and two dimensional random objects with conformal symmetries, such as planar Brownian motion, the Gaussian free field, the Schramm-Loewner evolution, and the conformal loop ensembles. In addition to their intrinsic beauty, these objects find applications in quantum field theory, statistical physics, and the theory of random surfaces. A guiding principle of the proposer's research is that many problems in stochastic geometry are best understood in terms of random surfaces and height functions (in discrete settings) and random distributions such as the Gaussian free field (in continuum settings).
Two dimensional random geometries are important in part because many problems in statistical physics (such as the way crystal surfaces fluctuate) are essentially two dimensional. Since the path-breaking work of Belavin, Polyakov, and Zamolodchikov in the 1970's and 1980's, it has been understood---at least heuristically---that the laws of certain macroscopic observables of these systems should be invariant under conformal maps from one planar domain to another. Over the past two decades, physicists have developed sophisticated non-rigorous techniques for understanding the properties of random objects with conformal symmetries. During the past few years, several mathematicians have begun to rigorously prove some of the predictions from the physics literature, along with many additional results. Many of the these problems have natural higher dimensional analogs that we are only beginning to understand. The investigator's proposed activities include efforts to bring new graduate students into this emerging field and to assist them in their studies and careers.
