Loewner Evolutions and Random Maps

University Of Washington, Seattle WA

Investigators

Abstract

Rohde will continue his investigations of conformal mappings generated by the Loewner differential equation, and will investigate the conformal geometry of random surfaces, particularly random planar maps such as the Benjamini-Schramm Uniformly Infinite Planar Triangulation. One aspect of Rohde's research is to understand similarities and differences between the deterministic and the stochastic Loewner equation, such as trace properties and their dependence on the regularity of the driving term. The Loewner equation is closely related to the "Zipper-algorithm" of Kuhnau and Marshall, and to conformal welding. Another aspect of Rohde's research is to prove convergence of this welding algorithm, and to study generalized weldings corresponding to non-simple curves. Rohde's research also explores the analytic foundations of the emerging theory of random maps and random Riemann surfaces, such as the type problem. What is the shape of a large molecule such as DNA? How does water percolate through soil? What are the patterns of spin-alignments in ferro-magnets? These problems are too complex to allow for a precise and simple mathematical answer. Even much simpler mathematical models, proposed by chemists and physicists, were out of reach of mathematicians for decades. This changed in 1999 with the invention of the Schramm-Loewner Evolution SLE, the Loewner equation driven by one-dimensional Brownian motion. The last decade has witnessed tremendous progress on old mathematical problems and even provided physicists with new and unexpected insights, as well as furthering interactions between probabilists, mathematical physicists, and complex analysts. Rohde's research aims at foundational questions of this emerging theory. Specifically, he is working on path properties of solutions of the Loewner equation in both the deterministic and probabilistic setting, and he is working towards an understanding of the large-scale behavior of random sphere-triangulations.