Conformal Maps and Harmonic Measure

共形图和谐波测量

• 批准号: 9800714
• 负责人: Nikolai Makarov
• 金额: $ 10.49万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800714&HistoricalAwards=false
• 关键词: Conformal; Maps; Harmonic; Measure

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.

提案：DMS-9800714首席研究员：尼古拉Makarov摘要：该项目的目标是推进知识在一些著名领域的复杂的分析，包括问题的保形焊接，扩散限制聚集，和遍历理论的调和测度的Julia集。共形映射和调和测度是研究二维物体的主要和经典工具。它们在科学和工程领域已经使用了很多年。最近，有相当大的兴趣在扩展的经典理论，以某些类别的集，具有复杂的（“分形”，“混沌”）结构，通常出现在动力系统或随机对象。一些最有趣的例子包括所谓的“拉普拉斯簇”，某些随机聚集体的增长率取决于调和测度。Laplacian集群作为数学模型的大量的自然现象，如扩散限制聚集的过程中，最近出现的晶体，粘性指，和介电击穿的形成模型。在物理学文献中，拉普拉斯簇已经被广泛研究。本计画将探讨这个主题，沿着其他一些基本的公开问题，涉及共形映射与调和测度的一般理论，以及在动态与随机分形上的应用。