Loewner Energy and Conformal Welding in Complex Analysis

这个研究项目关注的是自相似结构的几何。 自相似集是在不同尺度下看起来完全相同的集合，例如Koch雪花分形曲线。 更灵活的共形自相似概念要求一个集合在不同尺度下看起来大致相同，允许在每个尺度上有界失真的变化，例如在Julia集合中看到的。在自然界中观察到的大多数集合都是由随机过程产生的，并且满足更弱的“随机自相似性”概念：集合背后的随机机制在所有尺度上都是相同的。 这个项目旨在了解随机自相似集的基本性质，特别是与我们的共形自相似集的知识有关。 在这个项目中，树枝状晶体扮演着核心角色，而如何用保角坐标变换来描述这种分支随机分形的问题也是关键。Loewner微分方程是复分析中的一个基本工具，它提供了二维（平面）非自相交曲线（如Jordan曲线）和实值驱动函数之间的双射。 最近引入的“Loewner能量”的约旦曲线可以定义为Dirichlet能量的驱动函数。它先验地依赖于曲线的初始点。 在这个项目中，将研究与有限能量曲线的正则性有关的几个问题，旨在对能量进行更多的几何定义，并证明能量与初始点无关。 由于有限能量的曲线是拟共形曲线，因此将用拟共形分析的方法，特别是全纯变形和共形焊接来处理这些问题。 保形焊接的概括导致通过叠层枝晶的描述。 这一概括将探讨在确定性和随机设置。

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.