Mathematical Sciences: The Mapping Class Group and the Geometry of Teichmuller Space

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9503449 Masur This proposal concerns the geometry of Teichmuller space and the mapping class group. The proposer will use methods from complex geometry, probability, and dynamical systems to study Teichmuller metrics. In particular, the proposer wishes to consider random walks on Teichmuller space to study rigidity problems about the mapping class group. Teichmuller space represents the totality of pretzel-like surfaces with a fixed topological type; these surfaces, so called Riemann surfaces of a fixed genus, are crucial in the study of complex-valued functions as they generalize the complex number field. And the proposed research seeks to understand Teichmuller space from a geometric and metric point of view.