Colloquium abstracts

Sabyasachi Mukherjee*
TIFR, Mumbai
May 28, 2020

A New Link between Rational Dynamics and Kleinian Groups:  In the 1980s, Sullivan introduced an intriguing dictionary between two branches of conformal dynamics: actions of Kleinian groups and dynamics of rational maps. Although the dictionary has been an inspiration for translation of results and proof techniques from one world to the other, the translation is almost never automatic. The goal of this talk is to describe a direct topological/dynamical connection between Kleinian reflection groups arising from circle packings (e.g. the Apollonian gasket reflection group) and anti-holomorphic rational maps. In particular, the limit sets of these reflection groups are homeomorphic to the Julia sets of the corresponding rational maps via dynamically natural maps. We will also discuss examples of "conformal matings" of Kleinian reflection groups and rational maps. Time permitting, we will mention connections between such matings and classical questions in complex analysis such as conformal welding, coefficient bounds for univalent functions, etc