Mathematics Colloquium, Fall 2009
Gregory F. Lawler
University of Chicago
Random Laplacian Motion
I will discuss random curves whose evolution involves the solution of the Laplace equation with boundary values on the given curve. (This is an example of a "moving boundary" problem.) At the discrete level it is easy to define such models but it is hard to analyze them. At the continuum level, it is not obvious (or even known in some cases) how to give a mathematically precise definition of the curve. The interesting cases give irregular curves of nontrivial fractal dimension. I will survey some of the area discussing:
- Discrete case: Laplacian random walks and the loop-erased walk
- The role of dimension and why dimensions four and greater are not so interesting for this problem.
- Continuous case in two dimensions: We know much more than we did ten years ago because of the invention of the Schramm-Loewner evolution (SLE) by the late Oded Schramm. Wendelin Werner was awarded a Fields Medal in 2006 primarily for work on SLE and related problems. In two dimensions, conformal invariance plays a very strong role.
- The very open case of three dimensions.
- Speculations about the self-avoiding walk and its limits.
This talk is intended for a general mathematical audience.