Character Varieties and Locally Homogeneous Geometric Structures

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Studying all of the possible shapes of a geometric or mechanical object is a fundamental part of many problems in science and engineering, from understanding the folding of proteins or the formation of galaxies to programming autonomous vehicles to navigate complex terrain. This research project will broaden our understanding of a class of such "shape space" problems in which the geometric objects are surfaces (i.e. flat or curved two-dimensional shapes) or closely related spaces built from higher-dimensional pieces assembled in a two-dimensional pattern. These are natural examples to study because of the frequent appearance of surface geometry in a wide variety of mathematical problems and applications. In addition to contributing to mathematical knowledge, the computational and visualization components of this project will produce striking images of mathematical objects that exhibit their intricate structure and complexity in a way that can be appreciated by scientists and non-scientists alike.A locally homogeneous geometric structure on a compact manifold determines a holonomy representation, which is a homomorphism from the fundamental group of the manifold into a Lie group. The resulting map from the space of geometric structures to the space of group representations is always a local homeomorphism, but apart from a few classical examples (such as constant curvature geometries), the global behavior of the holonomy correspondence is not well understood. The investigator will contribute to our understanding of this basic problem by studying cases in which analytic methods can be brought to bear. For example, families of Anosov representations of surface groups in complex Lie groups are amenable to study through Kodaira-Spencer deformation theory, and through the application of tools from complex-analytic Teichmueller theory. In other cases, such as real projective structures on compact surfaces and generalizations thereof, geometric structures can be understood in terms of the solutions of a system of partial differential equations on the underlying manifold, allowing geometric questions to be attacked through techniques such as asymptotic analysis, barriers, and maximum principles.