Long plans to work on a wide spectrum of projects which explore a range of geometrical, topological and number-theoretical problems. Most of the issues to be explored are inspired by low dimensional topology, although several of the problems fit into a much wider context. These include the behavior of the spectral theory and homology of covering spaces, the structure of commensurators, the arithmetic of trace fields, the deformations and limits of real projective structures, and the role of similarity interval exchange maps in geometrically infinite surface groups. The unifying theme is the tying together of the many disparate aspects of hyperbolic manifolds in both low and higher dimensions. For while Perelman's work has drawn together the topological and geometric, our understanding of this geometry still has a long way to go. Progress in the directions proposed in this project would add significantly to this understanding.
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. Mathematicians understand how to interpret "like" very precisely and there are two very different notions which are of great importance in this setting: "topologically-like" and "geometrically-like". The recent work of Perelmann has verified a long-standing conjecture which emerged in the seventies with the work of Thurston, namely that the topological and geometrical pictures are intimately related. This is an important piece of global understanding. Amongst geometrical manifolds, by far the most important are the class called hyperbolic manifolds. This class is ubiquitous in many areas of mathematics, ranging from low-dimensional topology, to dynamical systems, to number theory. Indeed, this class is also crucially important in physics. However, even with Perlemann's work, hyperbolic manifolds themselves are still fairly poorly understood, although their importance have made them a magnet for research for well over thirty years. This proposal directs itself towards aspects of the structural study of hyperbolic manifolds and many issues related to them. For example, he intends to continue work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.
