The investigator plans to study spaces of hyperbolic metrics on a fixed 3-manifold. Unlike on a closed manifold, on an open 3-manifold there will typically be a large space of hyperbolic metrics. The investigators goal is to understand this space. Some questions are simple to state: Is the deformation space the closure of its interior? A more detailed description of the space is the ending lamination conjecture, which gives invariants that classify every hyperbolic metric on the manifold. This classification is not a parameterization; the map to the space of invariants is not continuous. The investigator is interested in understanding these discontinuities and describing the topology of the deformation space of metrics. Hyperbolic cone-metrics are a key tool used throughout this work. The philosophy is that very complicated geometry in a smooth hyperbolic structure can be exchanged for a less complicated, but singular, hyperbolic cone metric.
Three manifolds are mathematical spaces that locally look like the universe we live in. Mathematicians, beginning with Poincare, have been interested in classification question about three manifolds. Hyperbolic geometry is also a very old field dating back to the middle of the eighteenth century. In the last twenty-five years, largely due to the work Thurston, it has been realized that there are deep and beautiful connections between the two topics.
