Phases and Deformations of Loewner Evolutions

首席调查员将研究洛夫纳微分方程的性质。更具体地说，通过保角映射，Loewner方程提供了递增的二维集族和连续的一维函数(称为驱动函数)之间的对应关系。这种通信的性质没有得到很好的理解，是几个问题的来源。例如，驱动函数的性质如何影响二维增长？如何将二维几何图形转换为一维数据？这项工作将通过研究驱动函数的变形如何影响相应的几何来解决这些问题。这项研究的另一个目标是建立在最近涉及Loewner方程的令人兴奋的概率工作的基础上，并探索可以从当前的概率理解中恢复的确定性性质。Loewner微分方程是复数分析中的一个长期存在的工具。在过去的十年里，Loewner方程的一种新用途引起了人们的注意，不仅在复分析中，而且在概率和理论物理中也引起了注意：Loewner方程是被称为Schramm-Loewner演化(SLE)的随机过程的组成部分。这些过程由O.Schramm在2000年提出，已经成为解决数学家和理论物理学家都感兴趣的几个公开问题的工具。这项研究将加深对洛夫纳方程的理解，并将进一步推动这一合作领域的工作。此外，该项目还将促进本科生培训和指导方面的人力资源开发，目的是鼓励任职人数不足的群体更广泛地参与。

The principle investigator will study properties of the Loewner differential equation. More specifically, via conformal maps the Loewner equation provides a correspondence between increasing families of 2-dimensional sets and continuous 1-dimensional functions (called driving functions). The nature of this correspondence is not well understood and is the source of several questions. How, for example, do properties of the driving functions affect the 2-dimensional growth? And how is 2-dimensional geometry converted into 1-dimensional data? One of the ways in which this work will address these questions is by studying how deformations of the driving function affect the corresponding geometry. A further goal of this research is to build on the exciting recent probabilistic work involving the Loewner equation and to explore deterministic properties that can be recovered from the current probabilistic understanding.The Loewner differential equation is a long-standing tool in complex analysis. In the past ten years, a new use of the Loewner equation has attracted attention not only inside complex analysis, but also in probability and theoretical physics: the Loewner equation is an integral part of the random processes called Schramm-Loewner Evolution (SLE). Introduced by O. Schramm in 2000, these processes have been a tool in the solution of several open problems of interest to both mathematicians and theoretical physicists. This research will lead to a deeper comprehension of the Loewner equation and will further the work in this collaborative field. In addition, this project will contribute to human resource development in the training and mentoring of undergraduate students, with a goal of encouraging a broader participation of under-represented groups.