Geometry and dynamics of moduli spaces and mapping class groups of surfaces

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The theme of this project is to study the geometry and dynamics of flows on moduli spaces of hyperbolic surfaces, and actions of the mapping class group on representation varieties. Many questions discussed in this proposal and the methods applied are motivated by analogous statements on negatively curved manifolds and homogeneous spaces. Some of the problems to be addressed within the project are the following: 1) understanding the dynamics of the horocycle and earthquake flow on bundles over the moduli space. We aim to study the analogies between theses flows and unipotent flows on homogeneous spaces; 2) geometric properties of the space of Hitchin representations of surface groups in SL(n,R); and 3) Ergodic properties of the actions of the mapping class group on the representations of surface groups in to a compact complex lie group.The main goal of this proposal is to further explore and understand different geometric structures on surfaces and natural induced dynamical systems on the corresponding moduli spaces. The questions and methods in this project might provide new insights and results about deformations of geometric structures on surfaces, and the relationship between different models for generating random Riemann surfaces.