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Abstract: 

The classical theory of holomorphic dynamics includes the theory of Kleinian groups and the iteration theory of rational maps. In the 1980s Sullivan pointed out several analogies between these two subjects. For example, the concept of a Julia set of a rational map corresponds to the concept of a limit set of a Kleinian group. Even if one goes beyond holomorphic dynamics, an "extended" Sullivan dictionary can be very helpful as a guide for investigations. In my talk I will give an introduction to this subject and will discuss some recent developments.

Introduction to Speaker: 

Mario Bonk received his PhD from the Technical University of Braunschweig in Germany in 1988. Before moving to UCLA in 2010, he was a professor in the University of Michigan. He is currently Chair of the Department of Mathematics in UCLA and a Fellow of the American Mathematical Society.

Bonk's research interests lie at the interface of geometry and analysis, including classical complex analysis, the geometry of negatively curved spaces, dynamics of rational maps, and analysis on metric spaces. His current work often relies on an extension of classical results in geometry and analysis to a non-smooth or fractal setting. He has co-authored a research monograph on “Expanding Thurston maps” and written about 60 research articles. He was an invited speaker at the International Congress of Mathematicians (ICM) in Madrid in 2006.