A thin rubber sphere can be stretched, pushed in, and otherwise distorted so that its roundness is lost. For instance, one can contract the equator to transform the round sphere into an hourglass. While the neck of the hourglass does not look much like a round sphere anymore, the two bulbs are still pretty round. The celebrated Gauss-Bonnet theorem says that in some precise sense, no matter how the sphere is distorted, the average geometry of the result will still be round. Mathematicians view this result as an example where the topology of an object (i.e., the fact that it was produced by distorting a round sphere) constrains its geometry. The broad theme of this proposal is to develop similar results pertaining to higher dimensional abstract geometric objects.
Thurston's Geometrization Conjecture, proved by Perelman in 2003, states that every closed three-manifold can be cut into pieces, each of which admits one of eight types of homogenous metrics. Of these pieces, only the hyperbolic three-manifolds have not been classified. The full statement of the Perelman's result includes topological conditions that characterize when a closed three-manifold M admits a hyperbolic metric. Mostow's Rigidity Theorem implies that a hyperbolic metric on such an M is unique, if it exists, so it is natural to try to extract concrete geometric information about the metric from the topology of M. Effective geometrization, studied by the PI and his collaborators, is a program to extract concrete geometric information about a hyperbolic metric on a three-dimensional manifold from its topology. A large part of this project involves finding a description of the geometry of M given only that the number of elements needed to generate the fundamental group of M is bounded, or that M has a topological decomposition with certain characteristics. The key technique is to understand a sequence of closed hyperbolic three-manifolds asymptotically by passing to geometric limits. This perspective will also be applied outside of hyperbolic geometry, in the study of the growth of Betti numbers in sequences of higher rank locally symmetric spaces, in a program inspired by an active field in graph theory.
