Mathematical Sciences: Harmonic Measure, Conformal Mappings, and Geometric Measure Theory

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ABSTRACT DMS-9706875 Smirnov Smirnov will work on harmonic measure and distortion properties of conformal mappings. Harmonic measure is a fundamental object in complex analysis, providing a good description of the geometry of sets in the complex plane. This project will focus on the investigations into the fine (local) structure of harmonic measure, related properties of Green's functions, and possible applications to holomorphic dynamical systems. It seems that extremal behavior of harmonic measure occurs for fractal examples arising from complex dynamics and self-similar constructions. Therefore, powerful machinery, developed for investigations of dynamical systems (ergodic theory, thermodynamical formalism, holomorphic motions) might be useful (and perhaps necessary) for solving some problems in classical complex analysis. This project emphasizes research in, and interaction between two important areas of mathematics: classical complex analysis and dynamical systems. Dynamical systems is one of the more active areas of modern mathematics with applications to biology, engineering and physics. Classically, the analytical approach helped to solve many hard problems in dynamical systems. Recently it has become apparent, that answers to many classical analytical problems have a deep, dynamical origin.