Conformal Maps and Harmonic Measure

摘要：该项目的目标是推进复杂分析中一些知名领域的知识，包括保形焊接问题、扩散有限聚集问题和Julia集上谐波测度的遍历理论。保角映射和谐波测度是研究二维物体的主要和经典工具。它们已在科学和工程中使用多年。近年来，人们对将经典理论扩展到具有复杂（“分形”、“混沌”）结构的集合的某些类别产生了相当大的兴趣，这些集合通常出现在动力系统中或作为随机对象。一些最有趣的例子包括所谓的“拉普拉斯簇”，这是一种随机聚集，其增长速度取决于谐波测度。拉普拉斯簇是许多自然现象的数学模型，如扩散有限聚集的过程，最近作为晶体形成、粘性指指和介电击穿的模型而浮出水面。拉普拉斯聚类在物理文献中得到了广泛的研究。本项目将探讨这一主题，以及其他一些关于共形映射和调和测度的一般理论的基本开放问题，以及在动态和随机分形中的应用。

Proposal: DMS-9800714Principal Investigator: Nikolai MakarovAbstract: The goal of the project is to advance knowledge in some well-known areas of complex analysis, including the problems of conformal welding, diffusion limited aggregation, and the ergodic theory of harmonic measure on Julia sets. Conformal mappings and harmonic measure are major and classical tools in the study of two-dimensional objects. They have been used in science and engineering for many years. Recently, there has been considerable interest in the extension of the classical theory to some classes of sets which have complicated ("fractal", "chaotic") structure and typically arise in dynamical systems or as random objects. Some of the most interesting examples include so-called "Laplacian clusters," certain random aggregates whose rate of growth depends on harmonic measure. Laplacian clusters serve as mathematical models for a large number of natural phenomena such as the process of diffusion limited aggregation, which has surfaced recently as a model for the formation of crystals, viscous fingering, and dielectric breakdown. Laplacian clusters have been extensively studied in the physical literature. The project will explore this topic, along with some other basic open problems concerning the general theory of conformal mappings and harmonic measure, as well as applications to dynamical and random fractals.