Conformal Maps and Harmonic Measure

建议：DMS-9800714首席研究员：Nikolai Makarov摘要：该项目的目标是促进一些众所周知的复杂分析领域的知识，包括保角焊接、扩散限制聚集问题和Julia集上的调和测量的遍历理论。共形映射和调和测度是研究二维物体的主要和经典工具。它们已经在科学和工程中使用了很多年。近年来，人们对将经典理论推广到某些具有复杂(“分形”、“混沌”)结构且通常出现在动力系统中或作为随机对象的集合产生了相当大的兴趣。一些最有趣的例子包括所谓的“拉普拉斯星团”，即某些随机聚集体，其增长速度取决于调和度量。拉普拉斯团簇是大量自然现象的数学模型，如扩散限制聚集过程，最近作为晶体形成、粘性指进和介电击穿的模型浮出水面。拉普拉斯星团在物理文献中得到了广泛的研究。该项目将探索这一主题，以及其他一些基本的开放问题，涉及共形映射和调和测度的一般理论，以及对动力学和随机分形的应用。

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.