Deformation spaces of geometric structures

The PI will investigate deformation spaces of geometric structures on n-dimensional manifolds. A manifold is a mathematical object which looks locally like ordinary Euclidean space, but can have higher or lower dimension. Two-dimensional manifolds are called surfaces (e.g., the surface of a football, a donut or a pretzel), while our universe is an example of a 3-dimensional manifold. The study of deformations of possible geometric structures on manifolds arises naturally in many fields of mathematics as well as in physics. The PI will also contribute to the mathematical community through involvement in the Inquiry Based Learning Center at the University of Michigan, his involvement in the formation of the MACSS scholars program which is designed to support low-income students majoring in Mathematics, Computer Science and Statistics, curriculum development for undergraduate courses, serving as editor of mathematical journals, organizing conferences, and mentoring undergraduate students, graduate students and postdoctoral assistant professors. Specifically, the project focuses on the study of the Hitchin component of higher rank structures on surfaces and on space of hyperbolic structures on 3-manifolds. The Hitchin component is the pre-eminent example of a Higher Teichmuller space of representations of a closed surface group into a semi-simple Lie group. Striking analogies with classical Teichmuller theory have been discovered, including pressure metrics which generalize the Weil-Petersson metric from classical Teichmuller theory. The metric completion of the Teichmuller space of a closed surface, with respect to the Weil-Petersson metric, is the augmented Teichmuller space, which one may view as the orbifold universal cover of moduli space. PI will study the metric completion of the Hitchin component with respect to a pressure metric and to develop a geometric theory of the augmented Hitchin component which parallels the classical theory. The proof of Thurston’s Ending Lamination Conjecture developed many new tools for the study of hyperbolic 3-manifolds and their deformation spaces. In the original proof of the Ending Lam- ination Conjecture one obtains uniform bilipschitz models for hyperbolic 3-manifolds homotopy equivalent to an orientable closed surface. PI will develop uniformly bilipschitz combinatorial models for any hyperbolic 3-manifold with finitely generated, freely indecomposable fundamental group. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.