Loewner Evolutions and Quasiconformal Mappings

University Of Washington, Seattle WA

Investigators

Abstract

The principal investigator will study conformal mappings generated by the Loewner differential equation, and random conformal and quasiconformal mappings. The Loewner equation relates a continuously increasing sequence of planar simply connected domains to a real-valued function, the driving term of the equation. The correspondence is by means of conformal maps onto a standard domain (such as a disc). Through this mechanism complicated two-dimensional shapes can be encoded by seemingly simpler objects, namely, real-valued functions of a real variable. The correspondence between a shape and its driving term is complicated and leaves many open questions. The aim of this project is to provide a better understanding of this correspondence. For instance, the principal investigator will study the continuity of the sets under deformations of the driving terms. In light of Oded Schramm's SLE and the spectacular work of Lawler, Schramm, Werner, Smirnov and others, random driving terms (in particular, Brownian motion) are especially interesting and will be a focus of the research. Conformal mappings are often used to change coordinates from one region to a simpler region, such as a disc. They have applications in many areas within mathematics and to several branches of physics. On small scale, conformal maps look like rotations and dilations. Hence it is plausible that rotation- and dilation-invariant mathematical models of physical phenomena (e.g., Brownian motion, percolation, crystal growth, electrodeposition) are invariant under conformal coordinate changes. Theoretical physicists have long used this heuristic and obtained predictions for many of these models. Oded Schramm's discovery of the stochastic Loewner evolution (SLE, the Loewner equation driven by one-dimensional Brownian motion) and Smirnov's work on percolation have put this philosophy on a firm mathematical basis. The results obtained in recent years have generated a lot of excitement in both the mathematics and the physics communities. They have also created a new bridge between the two disciplines. A goal of this research is to shed new light on the mathematical side of this emerging theory.