Loewner Energy and Conformal Welding in Complex Analysis

复杂分析中的 Loewner 能量和保形焊接

基本信息

批准号：
1700069
负责人：
Steffen Rohde
金额：
$ 19.2万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700069&HistoricalAwards=false
关键词：
Loewner Energy Conformal Welding Complex

项目摘要

This research project is concerned with the geometry of self-similar structures. Self-similar sets are sets that look exactly the same at different scales, such as the Koch snowflake fractal curve. The more flexible notion of conformal self-similarity requires a set only to look roughly the same at different scales, allowing for changes of bounded distortion at each scale, such as is seen in Julia sets. Most sets observed in nature arise from processes involving randomness and satisfy the even weaker notion of "stochastic self-similarity": the random mechanism behind the set is the same at all scales. This project aims at an understanding of basic properties of stochastically self-similar sets, particularly in relation to our knowledge of conformally self-similar sets. A central role in the project is played by dendrites and the question of how such branching random fractals can be described using angle-preserving coordinate changes.The Loewner differential equation is a basic tool in complex analysis that provides a bijection between two-dimensional (planar) non-self-crossing curves (such as Jordan curves) and real-valued driving functions. The recently introduced "Loewner energy" of a Jordan curve can be defined as the Dirichlet energy of the driving function. It depends a priori on the initial point of the curve. During this project several questions related to the regularity properties of curves of finite energy will be investigated, aiming at a more geometric definition of the energy and a proof of the independence of energy from the initial point. Since curves of finite energy are known to be quasiconformal curves, the questions will be approached by methods from quasiconformal analysis, particularly holomorphic deformations and conformal welding. A generalization of conformal welding leads to a description of dendrites via laminations. This generalization will be explored both in the deterministic and in the stochastic setting.

该研究项目与自相似结构的几何形状有关。自相似的集合是在不同尺度（例如Koch雪花分形曲线）上看起来完全相同的集合。共形自相似性的更灵活的概念需要一个集合在不同的尺度上看起来大致相同，从而可以在每个尺度上改变有限失真的变化，例如在朱莉娅集合中看到的。自然界观察到的大多数集合源于涉及随机性并满足“随机自相似性”的较弱概念的过程：集合背后的随机机制在所有尺度上都是相同的。该项目的目的是了解随机上自相似集的基本特性，尤其是与我们对共同相似集合的知识有关。树突扮演的核心作用是通过树突扮演的问题，以及如何使用角度保护坐标坐标更改来描述这种分支随机分形的问题。LOEWNER微分方程是复杂分析中的基本工具，可提供二维（平面）非seff-Cross-Crosss-Crossing曲线（例如Jordan Curves）和真实价值的驾驶功能。 Jordan曲线最近引入的“ Loewner Energy”可以定义为驾驶功能的Dirichlet能量。这取决于曲线的初始点。在该项目中，将研究与有限能量曲线的规则性特性有关的几个问题，旨在对能量的更几何定义和能量独立性从初始点开始。由于已知有限能量的曲线是准文献曲线，因此将通过准文献分析（尤其是霍明型变形和保形焊接）的方法来解决这些问题。共形焊接的概括导致通过层压对树突的描述。将在确定性和随机环境中探索这种概括。