Conformal Mappings and Loewner Evoluation

University Of Washington, Seattle WA

Investigators

Abstract

Proposal Number: DMS-0201435 PI: Donald Marshall and Steffen Rohde ABSTRACT Conformal mapping and Loewner evolutions Marshall and Rohde will 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 decreasing sequence of simply connected planar domains. It relates a sequence of domains to a real-valued function, the driving term of the equation. Schramm's recent discovery of the stochastic Loewner evolution SLE, the Loewner equation driven by one-dimensional Brownian motion, has opened up a new area of investigations involving conformal mappings, probability theory and mathematical physics. The Loewner equation is also related to an algorithm for numerical conformal mapping. Conformal mappings have applications in many areas, both within and outside of mathematics, such as control theory, heat conduction, fluid dynamics, and complex dynamics. They are often used to change coordinates from one region to a simpler region like a disc. Regions with smooth boundaries are well understood. However, the appearance of fractals in many branches of science led to the natural problem of investigating regions bounded by highly nonsmooth, fractal curves, from the conformal mapping point of view. In recent years, fractal curves generated by random processes arising, for instance, in statistical physics, have received enormous attention. The core of Marshall's and Rohde's research is to better understand random fractal curves by means of conformal mappings, and conversely to study some problems about conformal mappings by analyzing domains bounded by fractal curves.