Hyperbolic geometry in dimensions 2 and 3

Two and three-dimensional spaces are fundamental objects lying at the intersection of many branches of mathematics. One approach to studying these spaces is to endow them with geometric structure and then use the geometry to study the spaces. The three most familiar geometries are the classical Euclidean or flat geometry, and the two non-Euclidean geometries: spherical and hyperbolic. It has been known since the late 1800s that most two-dimensional spaces (or surfaces) exhibit hyperbolic geometry. Thurston's groundbreaking work in the 1970s showed that "most" three dimensional spaces (or three-manifolds) also have hyperbolic geometry. The rich landscape of hyperbolic manifolds in two and three dimensions motivates this project's focus on hyperbolic geometry. The project will include many sub-projects suitable for training graduate students to work in this area. The PI, together with graduate students, will study questions around the geometry of hyperbolic 3-manifolds. One central topic will be the "renormalized volume" of a hyperbolic 3-manifold, and the connection between the Weil-Petersson gradient flow and the volume of the convex core. While renormalized volume has strong connections to physics, in this project the focus will be mathematical. The PI will study a version of renormalized volume of the universal Teichmüller space, and Thurston's skinning map. In another series of projects, the PI will study a family of curve complexes that interpolates between the usual curve graph and a quasi-tree, and will investigate the existence of actions of the mapping class group on median spaces and CAT(0) cube complexes. 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.