There has recently been a tremendous amount of progress in the areas of Kleinian groups and the geometry of 3-manifolds. Several long-standing conjectures have been solved, the most spectacular being Perelman's proof of the Geometrization Conjecture, but also including the Ending Lamination Conjecture (Minsky, Brock and Canary) and the Tameness Conjecture (Agol, Calegari-Gabai). The goal of this proposal is to get a more exact understanding of how geometric structures on 3-manifolds are reflected in their topological properties. Since hyperbolic structures on closed manifolds are unique, studying them through families typically requires allowing singularities or obtaining them by gluing flexible manifolds with boundary. Getting explicit, quantitative information about a geometric structure requires a refined understanding of the flexible structures from which they are obtained. These variational techniques can also be applied to understand other types of geometric structures, such as projective and Lorentzian structures, that don't have an invariant metric. We will also consider interesting questions about geometric structures in other dimensions (hyperbolic structures in high dimensions, projective structures on surfaces).
In this proposal we wish to study geometric structures in dimension 3, as well as other dimensions, using a combination of analytic, geometric, and topological tools. This is a very active and interesting area that has recently had an influx of new ideas and techniques. Geometry in dimension 3 is particularly appealing because it is visually accessible to many people, including beginning mathematics students and those with less technical backgrounds. It also has applications to physics and has led to the creation of a number of graphical interfaces that have been widely utilized.
