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的许多杰出猜想和预测。PI已经从事SLE研究多年，并在该领域做出了一些重要贡献。本课题主要研究各类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。该技术用于在单个域中以这样的方式构建多个SLE曲线，这些曲线彼此交换。应用它证明了Duplantier对偶猜想、弦性SLE和全平面SLE的可逆性（对于某些参数），以及在平面布朗运动上擦除环路得到SLE曲线是可能的。在建议的项目中，PI将继续应用耦合技术。这包括在Riemann球上构造简单的SLE回路，证明SLE自然参数化的可逆性，以及研究SLE在多个连通域中的行为。提出的研究计划将使人们更好地理解SLE曲线在单连通域和多连通域生长的行为。这将增加对不同类型域中许多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走）的理解，因为SLE已被确定为这些模型的缩放限制。该项目还将对共形场理论产生重大影响。这项建议的一个重要部分是其教育部分。PI将在密歇根州立大学开设概率和复杂分析本科课程，包括PI研究领域的最新发展。他还将教授SLE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.