Motivated by considerations in topology, geometry, arithmetic and group theory the proposal intends to study real, complex and quaternionic hyperbolic manifolds. This will involve the study of discrete groups, and their connections with other research active areas; for example number theory, and the theory of expanding graphs. We will also explore various interconnections between these geometric objects. This has been fruitful in the past, since previous work has shown how an understanding of higher dimensional hyperbolic manifolds can impact the topology of 3-dimensional manifolds.
The proposal seeks to further understand basic objects in modern geometry and topology, namely manifolds that admit a hyperbolic metric of some type. Many of the problems in the proposal are cross-disciplinary, and are amenable to study using a broad spectrum of mathematical technqiues; for example from number theory, geometry, topology and group theory. By their nature, progress and solutions to problems in the proposal will have a broader impact. The problems suggested have applications in fields as diverse as number theory (class numbers) and cosmology (and its relation to the topology of 4-dimensional hyperbolic manifolds) and computer science (expanding graphs). All of these have the potential to be fertile grounds for the education of graduate students.
