Abstract: A Thurston map is a branched covering map on a topological 2-sphere for which the forward orbit of each critical point under iteration is finite. Each such map gives rise to a fractal geometry on its underlying 2-sphere. The study of these maps and their associated fractal structures links diverse areas of mathematics such as dynamical systems, ergodic theory, classical conformal analysis, hyperbolic geometry, Teichmüller theory, and analysis on metric spaces. In my talk I will give an introduction to this subject and report on ongoing research.

Bio: 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.