Conformal Mapping

University Of Washington, Seattle WA

Investigators

Abstract

The Loewner differential equation was introduced in 1923 to study extremal problems for conformal maps in the unit disc. 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. It has led to the discovery of new results in percolation and random walks, for example, as well as the discovery mathematical proofs of results known to the theoretical physics community. The Loewner equation is also related to an algorithm for numerical conformal mapping discovered by Marshall and K\"uhnau. The Loewner equation has as input an arbitrary continuous function and produces a continuous family of conformal mappings. Marshall plans to investigate properties of the solutions of Loewner's equation under various assumptions on the smoothness of the driving function, and conversely to investigate how smoothness of the boundaries of the associated regions implies smoothness of the driving function. This is a classical problem where progress has been made only recently. Marshall will analyze convergence and error-estimates for the numerical mapping method called ``zipper'', closely related to Loewner's equation, and improve the speed of convergence using ``generational'' techniques. The main theme of Marshall's research program is to study conformal mappings generated by the Loewner differential equation, and related topics. Conformal mappings have been used as a tool in science and engineering for many years. They are often used to change coordinates from a complicated region to a simpler region like a disc. Physical processes are modeled by partial differential equations. A partial differential equation on the complicated two dimensional region can be changed by a conformal map to a similar equation on the disc, a setting where it is easier to solve. Classically, this method was used for problems related to Laplace's equation, such as electrostatics and two dimensional fluid flow. Numerous non-classical applications have been developed in the last three decades such as electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, and heat transfer. Fundamental research in conformal mapping and conformal field theory are important directions in mathematics and physics respectively. The fusion of Loewner's differential equation and probability forms a bridge between these two areas. This project is likely to increase the understanding of solutions to Loewner's equation, as foundational work, which should increase its usefulness in understanding stochastic processes. Broader impacts include the continued improvement and dissemination of the conformal mapping computer codes, which have been used by a number of investigators not in mathematics, as well as mathematicians. Greater speed and new knowledge of convergence should lead to wider applicability and use of this algorithm. Broader impacts also include helping to cement a stronger relationship between modern complex analysis and mathematical physics, which should benefit both.