Geometry of Teichmuller Space and Mapping Class Group

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Most of this research proposal is dedicated to the study of the geometry and topology of Teichmueller space, moduli space, and mapping class groups. Some specific projects involve solving some generalizations of the conjugacy problem for mapping class groups, investigating the properties of the systole-length function on moduli space, and finding a coarse description of the geodesics in the Thurston metric on Teichmueller space. The PI also proposes to study the homology growth of irreducible automorphisms of a free group in finite covers. Surfaces are two-dimensional spaces like the surface of a ball or a doughnut. Their topological and geometric properties have captured the minds of mathematicians for centuries. The study of surfaces is not only interesting and beautiful, but has generated entire fields of mathematics and arguably lies at the intersection of all fields of mathematics. One way to understand a surface is to study the various ?natural? geometric structures on the surface and how the different structures relate to each other. The parameter space all geometric structures on a given surface is a fundamental mathematical object called the moduli space. This is usually a higher-dimensional space that itself naturally carries many rich and interesting geometric structures. The study of moduli space is part of the guiding mathematical principle that, in order to understand one particular structure on a space, one must try to understand the structure of the meta space comprised of all structures. The work of the PI is dedicated to expanding our knowledge about moduli space and some related spaces.