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

Smirnov将研究共形映射的谐波测量和畸变特性。调和测度是复分析中的一个基本对象，它很好地描述了复平面上集合的几何形状。本项目将重点研究谐波测度的精细（局部）结构，格林函数的相关性质，以及在全纯动力系统中的可能应用。谐波测度的极值行为似乎出现在由复杂动力学和自相似结构引起的分形实例中。因此，为研究动力系统（遍历理论、热力学形式论、全纯运动）而开发的强大机制可能对解决经典复分析中的一些问题有用（也许是必要的）。本项目重点研究数学的两个重要领域：经典复分析和动力系统之间的相互作用。动力系统是现代数学中较为活跃的领域之一，在生物学、工程学和物理学中都有应用。经典的分析方法有助于解决动力系统中的许多难题。最近很明显，许多经典分析问题的答案都有深刻的动力学根源。

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.