Long plans to work on various old and new projects in geometric, algebraic and structural aspects of low and high dimensional hyperbolic manifolds. The problems are drawn from a variety of areas: One family of problems comes from questions about surface groups, which are of interest not only in low dimensional topology, but also in certain parts of number theory. As an example, we will investigate whether hyperbolic 3-orbifolds must contain a surface group, or whether one can increase the homology of a hyperbolic bundle. Another family of questions come from the study of character varieties. In higher dimensions the proposer will consider constructions as well as geometrical bordism questions coming from physics. Finally, the proposer will work on projects connected to a broad range of questions arising from the arithmetic of hyperbolic manifolds.
A space is called a 3-manifold if it is made of small chunks all of which are ``like'' the ordinary 3-dimensional space that we live in. This proposal directs itself towards aspects of the structural study of the most important class of 3-manifolds, the so-called hyperbolic 3-manifolds. This class is ubiquitous in mathematics and physics, for example although the nature of the universe has yet to be resolved, there is at least one group who have suggested that the universe is the Picard orbifold - a hyperbolic orbifold. The proposer has results which restrict the sorts of manifolds which can arise in certain physical models of the universe. He also intends to contain work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.
