Some remarks on SLE and an extended Sullivan dictionary

Abstract: The Sullivan dictionary translates statements about Kleinian groups into statements about Julia Sets and vice versa. For example, a limit set on the Kleinian group side corresponds to a Julia Set, and the orbit of a point under a Kleinian group corresponds to the inverse images of a point by a rational map. We will discuss adding another category to the dictionary, namely SLE, Schramm-Loewner Evolution. Here limit sets and Julia Sets correspond to the SLE “trace”. We will point out that with suitable modifications, the Sullivan dictionary can be enlarged to include SLE. As one example we will discuss the various analogues of the Ahlfors conjecture for Kleinian groups. Another subject we will discuss is the various versions of rigidity that appear in the dictionary. This lecture is aimed at a general mathematical audience and we will stress ideas, not technicalities appearing in proofs.