CAREER: Analysis of the Geometric Properties of the SLE Curves

由Oded Schramm于1999年提出的Schramm-Loewner演化(SLE)已被成功地用于证明统计物理学中许多杰出的猜想和预言。该协会多年来一直致力于系统性红斑狼疮的研究，并在这一领域做出了一些重要贡献。该项目的重点是研究各种SLE曲线的几何性质。为了完成这个项目，PI将使用他之前开发的随机耦合技术。这种技术被用来在单个区域中构造多条SLE曲线，使得这些曲线彼此可交换。它被用来证明DuPlantier的对偶猜想，弦SLE和全平面SLE(对于某些参数)的可逆性，以及可以在平面布朗运动上擦除环以得到SLE曲线。在拟议的项目中，PI将继续应用耦合技术。这包括在Riemann球面上构造简单的SLE环，证明SLE的自然参数化的可逆性，以及研究SLE在多连通区域上的行为。提出的研究计划将使人们更好地了解在单连通和多连通域中增长的SLE曲线的行为。这将增加对不同类型区域中的一些二维统计物理晶格模型(例如，临界渗流、临界伊辛模型、高斯自由场和环擦除随机游动)的理解，因为SLE已被确定为这些模型的标度极限。该项目还将对共形场理论产生重大影响。这项提议的一个重要部分是它的教育部分。PI将在密歇根州立大学开设概率和复数分析本科课程，以纳入PI研究领域的最新发展。他还将教授关于系统性红斑狼疮的专业研究生课程，并让研究生在他的指导下工作。

The Schramm-Loewner evolution (SLE) introduced by Oded Schramm in 1999 has been used with great success to prove many outstanding conjectures and predictions from Statistical Physics. The PI has been working on SLE for a number of years, and has made some important contributions to this area. The proposed project focuses on studying the geometric properties of the SLE curves of various kinds. To do this project, the PI will use the stochastic coupling technique he developed earlier. This technique is used to construct more than one SLE curve in single domain in such a way that the curves commute with each other. It was applied to prove the Duplantier's duality conjecture, the reversibility of chordal SLE and whole-plane SLE (for some parameters), and that it is possible to erase loops on a planar Brownian motion to get an SLE curve. In the proposed project, the PI will continue the application of the coupling technique. This includes constructing simple SLE loops on the Riemann sphere, proving the reversibility of the natural parametrization of SLE, and studying the behavior of SLE in multiply connected domains. The proposed research plan will give people better understanding of the behavior of the SLE curves growing in simply and multiply connected domains. This will then increase understanding of a number of two-dimensional Statistical Physics lattice models (e.g. critical percolation, critical Ising model, Gaussian free field, and loop-erased random walk) in different kinds of domains since SLE have been identified as the scaling limits of these models. The project will also have significant impacts on Conformal Field Theory. A significant part of this proposal is its educational component. The PI will develop undergraduate courses on Probability and Complex Analysis at Michigan State University to include the newest development of the PI's research area. He will also teach specialized graduate courses on SLE, and involve graduate students in working under his supervision.