Geometric Function Theory and Loewner Evolutions

University Of Washington, Seattle WA

Investigators

Abstract

Abstract Geometric function theory and Loewner evolutions Rohde will continue to investigate conformal mappings generated by the Loewner differential equation and related topics. The Loewner differential equation describes the flow associated with the conformal mappings onto a continuously increasing sequence of simply connected planar domains. It encodes such a sequence of domains into a real-valued function, the driving term of the equation. The recent discovery of the stochastic Loewner evolution SLE by Oded Schramm (the driving term is one-dimensional Brownian motion), together with results such as Smirnov's proof of convergence of critical percolation clusters to a SLE(6), has opened up a new area of investigations involving conformal mappings, probability theory and mathematical physics. The list of critical lattice processes from statistical physics that are conjectured to converge to SLE is constantly growing, and driving terms different from Brownian motion are under investigation. Self-similar sets (so called "fractals") such as the van Koch snowflake have played an important role in mathematics because they serve both as toy models for physical phenomena, and are ameanable to mathematical analysis (dynamics, ergodic theory, conformal mapppings). The more recent recurrent appearance of "random fractals" (sets that resemble fractals but are only statistically self-similar) in various branches of mathematics, probability and statistical physics necessitates a mathematical foundation allowing rigorous analysis. Schramm's SLE is one of the most exciting developments in this direction and has resulted in verifications of numerous predictions made earlier by physicists. The core of Rohde's research is to better understand random fractals and to answer several questions about them by means of conformal mappings.