Abstract: Many research level problems in mathematics require lots of technical looking definitions from many advanced classes just to state, but the goal of this talk is to discuss some easy-to-state open problems about random sets in the plane. Most of these problems are probably pretty hard: several have been attacked by Fields medalists without success. There will be lots of pictures, some numerical evidence, but not much in the way of proofs. I will start with an informal introduction to Brownian motion as the limit of discrete random walks, and define harmonic measure as the first hitting distribution of Brownian motion in a domain on the boundary. We will then discuss some known facts about Brownian paths in the plane and state some open problems about them, e.g., does a Brownian path cover a rectifiable path? We will end with a discussion of DLA (diffusion limited aggregation), a much studied random growth process about which almost nothing is rigorously known.
