13th Annual Thomas Wolff Memorial Lecture
April 1, 3 and 7, 2014
4:00 p.m. 151 Sloan
Random Walks and their Scaling Limits
The simple random walk on the integer lattice is well understood as well as its scaling limit, Brownian motion. However, there are a number of models of random walks with strong interactions for which we are still trying to determine the behavior. My first talk will be a survey of the state of knowledge for three problems: intersections of random walk, loop-erased random walk, and self-avoiding random walk.
There has been considerable work in the last twenty years on the planar case and I will spend the second two lectures discussing aspect of this. In the the second lecture, I will consider a particular case, the loop-erased random walk, and describe its relationship to other models in particular spanning trees, determinants of the Laplacian and the random walk loop measures. If time allows. I will discuss a recent result with C. Benes and F. Viklund about the probability that a loop-erased walk goes though a point.
The third talk will focus on the continuum limit of many of these walks, the Schramm-Loewner evolution (SLE). Many of the properties SLE can be seen as continuous analogues of properties of the loop-erased walk and I will discuss some of them including recent work on fractal properties with a number of coauthors.
Greg Lawler received his undergraduate degree from the University of Virginia in 1976 and his Ph.D. at Princeton University in 1979 under the direction of Edward Nelson. He has held faculty positions at Duke University and Cornell University and is currently the George Wells Beadle Distinguished Service Professor in mathematics and statistics at the University of Chicago. He is a member of the National Academy of Sciences and is a fellow of the American Academy of Arts and Sciences, American Mathematical Society, and Institute of Mathematical Statistics. He was an invited speaker at the 2002 International Congress of Mathematicians and was a co-recipient of the Polya prize from SIAM in 2006. He is the author or co-author of six books.
Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize
Caltechs great analyst who was tragically killed at age 46 in an automobile accident
in July 2000. Wolff was a specialist in
analysis, particularly harmonic analysis. Professor Wolff made numerous highly original
contributions to the mathematical fields of Fourier analysis, partial differential
equations, and complex analysis. A recurrent theme of his work was the application of
finite combinatorial ideas to infinite, continuous problems.
His early work on the Corona theorem, done as a Berkeley graduate student, stunned the mathematical community with its simplicity. Tom never wrote it up himself since several book writers asked for permission to include the proof in their books where it appeared not long after he discovered it. After producing a number of very significant papers between 1980 and 1995, he turned to the Kakeya problem and its significance in harmonic analysis, works whose impact is still being explored.
Peter Jones, professor of mathematics at Yale, described Tom’s contributions as follows: “The hallmark of his approach to research was to select a problem where the present tools of harmonic analysis were wholly inadequate for the task. After a period of extreme concentration, he would come up with a new technique, usually of astonishing originality. With this new technique and his well-known ability to handle great technical complications, the problem would be solved. After a few more problems in the area were resolved, the field would be changed forever. Tom would move on to an entirely new domain of research, and the rest of the analysis community would spend years trying to catch up. In the mathematical community, the common and rapid response to these breakthroughs was that they were seen not just as watershed events, but as lightning strikes that permanently altered the landscape.”
Tom Wolff was noted for his analytic prowess, the depth of his insights, and the passion with which he nurtured the talents of young mathematicians. We miss him.