Abstract: If two closed hyperbolic manifolds of dimension n ≥ 3 are homeomorphic, then they are isometric according to Mostow's celebrated rigidity theorem of 1968. One can argue that this landmark result was a major inspiration for many transformative developments in mathematics such as Gromov's theory of negatively curved spaces or the analysis on metric spaces as spearheaded by Heinonen and Koskela in the late 1990s. Another fruitful theme in this general area has been Sullivan's dictionary exhibiting strong analogies between rational dynamics and the theory of Kleinian groups. In my talk I will give a survey on this and related topics and address some recent developments.


Bio-Sketch: Mario Bonk  is a professor at University of California, Los Angeles (UCLA), where he recently finished his tenure as the Chair of the Department of Mathematics, and an elected fellow of the American Mathematical Society. He received his PhD from the Technical University of Braunschweig in Germany in 1988, and was a professor at University of Michigan before moving to UCLA. 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. He has co-authored a research monograph on "Expanding Thurston maps" and written about 60 research articles, publishing on Ann. Math., Invent. Math., etc. He was an invited speaker at the International Congress of Mathematicians (ICM) in Madrid in 2006.