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K-State Today

April 25, 2019

University of Tennessee, Knoxville professor to present mini-course 'Random Spanning Trees and SLE'

Submitted by Reta McDermott

Professor Joan Lind, University of Tennessee, Knoxville, will present a mini-course titled "Random Spanning Trees and SLE" May 6-10. Professors Nathan Albin and Pietro Poggi-Corradini will take a few hours on Monday, May 6, and Friday, May 10, to fill in some background material.

Course information:

Mini-course: Math 799 — Top/Math
Section: ZA
Number: 18300
Credits: One credit
Time: 5:30-7:55 p.m. Monday-Friday from May 6-10
Place: 204 Burt Hall 

Abstract: Schramm Loewner Evolution (SLE) consists of a rich family of random processes in the plane. At the heart is the Loewner differential equation, a 100-year old tool from geometric function theory. In the 90's, Oded Schramm combined the Loewner equation with Brownian motion to create SLE, and his new process became the key to unlocking many important questions in probability and physics.

The questions arising from physics involved random curves associated with lattice models such as percolation models, the Ising model, and more. In particular, one could ask what happens to these random curves as the lattice spacing goes to zero? In many cases, SLE answered this question of the existence of the scaling limit.

This course will showcase the world of SLE, which connects complex analysis, probability, discrete math, and physics. We will focus on one particular lattice model: the Peano curve associated with the uniform spanning tree. After introducing this model, the Loewner equation and SLE, we will consider the question of the scaling limit of the Peano curve. The main goal of this course is to answer this question and give a connection between uniform random spanning trees of planar grids and SLE(8), which is a process of random space-filling curves. This result is theorem of Lawler, Schramm, and Werner (Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), no. 1B, 939–995).

Time permitting the course will also explore the notion of fair random spanning trees and its connection to spanning tree modulus.