Geometry and dynamics on deformation spaces of geometric structures

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In his Erlanger program of 1872, Felix Klein defined geometry to be the study of the properties of a space that are invariant under its group of symmetries. It was Charles Ehresmann in 1935 who started the study of deformation spaces of geometric structures, asking which "shapes" can be "locally modeled" on a certain geometry. In 1982 William Thurston's Geometrization Conjecture, now a theorem, renewed interest in locally homogeneous spaces, that is, spaces that look the same at each point. This research project studies families of structures on manifolds and how they change when one perturbs them, focusing in particular on geometric and dynamical aspects. As a broader impact, the investigator will involve graduate students in her work, organize activities aimed at junior mathematicians, and deliver lectures outside the University. She also wants to promote the collaborative side of the work, and to create a supportive and attentive environment for members of groups underrepresented in mathematics.The investigator will employ results and techniques developed in the context of hyperbolic structures to study other geometric structures. For example, she will investigate compact or hyperideal convex polyhedra in anti-de Sitter space, a Lorentzian analogue of the hyperbolic space, and the end(s) of complex hyperbolic manifolds. Many deformation spaces arise from spaces of representations of the fundamental group of a manifold into a Lie group, so the PI is also planning to continue the study of the dynamical decomposition of character varieties of free groups, and of fundamental groups of hyperbolic manifolds with compressible boundary. Finally, the PI will study "higher Teichmueller theory", that is, representations of a surface group into Lie groups of higher real rank, and Anosov representations, which are a dynamical analogue of locally homogeneous geometric structures. Since Anosov representations turn out to be generalizations of convex cocompact subgroups of rank one Lie groups to the context of discrete subgroups of Lie groups of higher rank, the PI plans to use techniques developed for Kleinian groups in order to study limits of Anosov representations. It is anticipated that results and techniques coming from differential geometry and low-dimensional topology will inspire new research directions with deep connections with dynamical systems, Lie theory, complex analysis, and even algebraic geometry, number theory, representation theory, and physics.