CAREER: Geometric Structures, Character Varieties, and Higher Teichmuller Theory

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

In his Erlanger program of 1872, Felix Klein defined geometry to be the study of properties of a space which 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, thanks to Grigori Perelman, renewed the interest in locally homogeneous spaces, that is spaces that look the same at each point. The PI proposes to study 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 PI wants to develop a more inclusive environment for undergraduate students, graduate students, postdocs and early career mathematicians. She proposes to organize: a Directed Reading Program which pairs undergraduate students with graduate mentors for independent projects; a Women in Geometry and Topology network with a website, annual dinners at conferences and summer retreats where participants will start mutual collaborations; a Mid-Atlantic Math Alliance Program to build a regional community of mentors who will work with underrepresented minority students to help them succeed in their careers; Log Cabin Conferences gathering a small group of researchers (many early career) in a remote location to learn a new topic in a collaborative atmosphere. In addition, she plans to continue organize the Diversity Lecture Series, an annual Sonia Day for middle school girls, the Math Club for undergraduate students at UVa, to be faculty advisor for the AWM Student Chapter, mentor for the AWM program and for the Math Alliance program, to organize the Geometry Seminar, the Virginia Topology Conference and annual graduate reading courses. Hyperbolic structures are the prototypical example of geometric structures with interesting deformation spaces. The PI wants to use results and techniques developed in the context of hyperbolic structures for studying other geometric structures. For example, she plans to investigate analogue structures in anti-de Sitter space. A lot of 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 wants to study "higher Teichmueller theory," that is "nice'" 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. Differently from classical Teichmuller theory, it is not known, in general, if these representations are holonomies of geometric structures. The PI wants to study this question, together with the description of limits of these representations and a different topology on these spaces, the geometric topology. The PI thinks 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. 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.