Dynamics in Hyperbolic Geometry and Teichmuller Theory

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This project lies at the intersection of dynamics, geometry and topology. The project aims to study various dynamical questions in hyperbolic geometry and Teichmuller theory. The main focus will be to continue the study of random walks on groups acting on hyperbolic space and the mapping class group acting on Teichmuller space. The goal is to extend further the analogy between the two situations, specifically to show that the associated harmonic measures on the boundaries of these spaces share similar properties. Other projects include the study of dynamical aspects of non-classical interval exchanges and continuation of the study of translation lengths of pseudo-Anosov maps on the curve complex. The relationship between dilatations and curve complex translation lengths along a fibered face for the Thurston norm on the second homology of a fibered hyperbolic 3-manifold will also be explored.The theory of surfaces such as the surface of the ball or the donut, is of fundamental interest across many fields of mathematics. Teichmuller theory is the study of the possible shapes a surface can assume. The proposal focuses on these shapes and their interactions with certain transformations of the surface called mapping classes. The main goal is to explain how the shape evolves under random transformations. Other projects include understanding the complexity of the transformations. One measure of complexity is the complexity of curves obtained on the surface by repeating a transformation. The problems outlined in the proposal are important in varied fields like geometry, topology and dynamical systems. It is hoped that the proposal will initiate fruitful collaborations and offer mentoring opportunities at several levels from postdoctoral fellows to undergraduate students.