Geometric Structures on Surfaces

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The PI proposes to continue his research in geometric structures on low-dimensional manifolds, deformations of discrete groups, and moduli of surface group representations. One problem involves completing the classification problem for proper affine actions, a basic problem in geometry whose history stretches back to the classification of Euclidean crystallographic groups in the early 20th century. In the last 25 years, the classification in 3 dimensions has been reduced to a problem in 2-dimensional hyperbolic geometry. Another side of this general problem stems from the observation that deformation spaces of geometric structures are themselves locally modeled on representation spaces of fundamental groups. This naturally leads to dynamical questions on actions of mapping class groups. Moduli spaces of surface group representations support invariant Poisson structures, as well as finer structures depending on a conformal structure. The dependence of these as the Riemann surface varies over Teichmueller space is crucial to understanding dynamics of surface group representations.This research focuses on the relation between topology and geometry. While topology concerns the loose qualitative organization of patterns of points, geometry concerns quantitative measurements such as distance, angle and area. These two viewpoints are compared through symmetries, leading to algebraic calculations involving groups. The Experimental Geometry Lab has been a center for the PI and his students and collaborators (at all levels) to develop tools for these calculations and visualization. Many of the above projects have an experimental component and the PI will continue to work with students (from high school to postdocs) to develop user-friendly software to assist in their investigation and dissemination.