Special topics course proposal Title: Schramm-Loewner Evolutions and Two-Dimensional Statistical Physics Instructor: Tom Kennedy This course will be an introduction to one of the most spectacular recent discoveries at the interface of mathematics and physics - the Schramm-Loewner Evolution (SLE), which brings together ideas from probability, complex analysis, geometry and physics. SLE is a one parameter family of stochastic processes that produce random curves in the plane. The past few years have shown that these processes decribe the random curves that are found in a wide variety of two dimensional statistical physics systems at their critical point. "Critical points" are values of the parameters of the system with the property that the randomness of the system, which typically is seen only at microscopic length scales, produces random structures that can be seen at macroscopic length scales. Conformal invariance plays a key idea in the SLE process and its connection to statistical physics models. Roughly speaking conformal invariance means that the model is not changed by a map in the plane that preserves angles. There are lots of such maps in two dimensions, and the rich nature of this symmetry produced in the 1980's something called conformal field theory which revolutionized our understanding of two-dimensional critical phenomena. The results of the past few years on SLE are advancing our understanding of the geometry of two-dimensional critical phenomena in an equally exciting fashion. Some examples of two-dimensional models described by SLE include the self-avoiding walk, the loop-erased random walk, interfaces in critical percolation, and the frontier of Brownian motion. This course will start with an explanation of what all these models are. Then we will motivate and define the Schramm-Loewner evolution process, including some background in conformal maps (the Loewner equation) and stochastic differential equations. Next we will derive some of the properties of the SLE process. Two particular properties of SLE only hold for special values of the parameter, and this fact helps make the connection between particular cases of SLE and particular physics models. Finally we will take a look at some of the proofs that particular models are described by SLE. Text: We will roughly follow Greg Lawler's book "Conformally invariant processes in the plane" which may be downloaded from his website. Prerequisites: SLE involves conformal maps in the plane, Brownian motion and stochastic differential equations, but I will cover what we need to know (as does Lawler's book). I will assume an undergraduate knowledge of complex variables and probability.