Mathematical Sciences: The Topology and Geometry of Hyperbolic 3-Manifolds

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Professor Canary's work concerns the relationship between the geometry and the topology of hyperbolic 3-manifolds. In the past he has explored the consequences of knowing that a hyperbolic 3-manifold is topologically tame, i.e. homeomorphic to the interior of a compact 3-manifold. In particular, he studied the spectral geometry and the dynamics of topologically tame hyperbolic 3-manifolds. One application was a proof of Ahlfors' measure conjecture for topologically tame hyperbolic 3-manifolds. In this project he intends to study the question of which hyperbolic 3-manifolds are topologically tame. The conjecture is that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame. He also intends to study the space of all hyperbolic structures on a given 3-manifold. An n-manifold is an object such that about every point there is a neighborhood which looks like a ball in n-dimensional space. For example, the unit sphere (in 3-dimensional space) is a 2-manifold, since about each point there is a small neighborhood which looks like a disk in the plane. One natural way in which manifolds arise is as solution sets of equations in several variables. For example, the unit sphere is the solution set of a particularly simple equation (x^2 + y^2 + z^2 = 1). A metric on a manifold M is a way of measuring distances between points on M. A manifold may admit many different metrics. For example, the unit sphere and the surface of an apple are topologically the same manifold, but they have different metrics. Studying the types of possible metrics on a manifold has many applications in mathematics and physics. The investigator will study metrics on 3-manifolds such that about any point there is a small neighborhood which may be identified (by an identification which preserves distances) with a ball in hyperbolic 3-space. These metrics are called hyperbolic metrics. Hyperbolic metrics on 2-manifolds have recently been of great interest to physicists, in particular, to string theorists.