This proposal is motivated by the unification between the topology and the geometry of three-dimensional manifolds. It is primarily focused on the quantitative geometry of hyperbolic 3-manifolds, specifically on estimating such quantitative invariants as volume, Margulis number and diameter in terms of topological data. This is an area in which connections with topology and several other branches of mathematics are playing unexpected roles. Classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950s, become particularly powerful when applied in the context of hyperbolic geometry. These topological ideas interact with more geometric and analytic methods, such as the log(2k-1) Theorem of Anderson, Canary, Culler and Shalen; the isoperimetric inequality for hyperbolic space; the theory of algebraic and geometric convergence of Kleinian groups; the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary; and the work of Agol, Dunfield, Storm and Thurston which applies properties of the Ricci flow with surgeries to the study of Haken manifolds and Dehn filling. Furthermore, surprising interactions are emerging, via topology, between quantitative geometry of hyperbolic 3-manifolds and their number-theoretic aspects, specifically their trace fields; this has allowed applications of deep results in number theory to the subject.
Non-Euclidean geometry is a classical topic in pure mathematics which has seen remarkable developments in recent decades. The subject had its origin in the attempt, begun in ancient times, to prove that Euclid's fifth axiom could be deduced from his other axioms. It was shown in the course of the 19th century that this cannot be done: there is a mathematical structure called the hyperbolic plane (in two dimensions) or hyperbolic space (in three or more dimensions) which satisfies all of Euclid's axioms except the fifth, and in which the sum of the angles of a triangle is always less than 180 degrees. Remarkably, hyperbolic geometry turns out to be much richer than Euclidean geometry. This accounts for the astonishingly varied interactions that have developed since the 1960's between hyperbolic geometry and other branches of mathematics and science. Most of these interactions involve the study of hyperbolic manifolds, which are geometric objects that have the small-scale geometry of hyperbolic space but have a more complicated structure in the large. For example, a straight line in hyperbolic space, as in Euclidean space, always extends to infinity; but in a hyperbolic manifold, a path that is locally a straight line (called a geodesic) may exhibit globally "periodic" behavior like a circle. Some of the principal investigators' earliest work on hyperbolic manifolds produced a result about knots that has been applied to study the structure of recombinant DNA. They are at present investigating a variety of aspects of the geometry of hyperbolic manifolds and connections with some of the other topics mentioned above.
