In the theory of rational maps (holomorphic functions from $\mathbb{CP}^1$ to itself), a natural question is how to describe the maps. This is most tractable in the case when the map is \emph{post-critically finite}. In this case, we can describe the dynamical system in terms of a correspondence on graphs.

On the one hand, this correspondence on graphs allows us to characterize which topological maps from the sphere to itself can be made into a geometric rational map. On the other hand, these graph correspondences can be generalized to give short descriptions of self-similar groups, for instance a concise description of the Grigorchuk group, the first-constructed group of intermediate growth. Speaker(s): Dylan Thurston (Indiana University)