There are two problems suggested in this proposal for further investigation. The first problem is completing our understanding of the deformation space of hyperbolic structures on open 3-manifolds and describing the geometry of such structures. This problem has been the subject of many studies started by Bill Thurston and followed by Bonahon, Canary, Minsky, Brock and others. Their work provided very interesting results especially in the case of incompressible boundary 3-manifolds. However such a nice description does not exist in general for hyperbolic structures on other 3-manifolds and in particular handlebodies. Namazi will focus on deepening our understanding of this general case. The second problem deals with the geometry of closed hyperbolic 3-manifolds. A very fundamental problem in the subject is to describe the geometry of a hyperbolic manifold and find a recipe to construct a very accurate approximation for the hyperbolic metric. These two problems are very closely related to each other and in earlier work Namazi was able to construct models for the geometry of a large class of hyperbolic structures on handlebodies and this was used to construct accurate approximations of the hyperbolic metric on a family of closed 3-manifolds. The hope is to extend this to a broader set of closed 3-manifolds.
A 3-manifold is an object, which looks like the regular Euclidean 3-dimensional space around every point. The problem of classifying 3-manifolds has been one of the fundamental questions in mathematics, which perhaps started by Poincare's work and conjecture in the early twentieth century. Thurston pushed the subject much further and conjectured that understanding and classifying hyperbolic 3-manifolds results in a classification of all 3-manifolds. The recent development of the subject in the last few years has brought us closer than ever to see a complete proof of Thurston's conjecture. A consequence of this classification is the fact that these fundamental objects have an almost canonical shape. What remains unanswered is a description of this shape and this is the major subject that Namazi will try to address in this project. Similar to his earlier work he also expects that such a description makes it possible to answer many unanswered questions in the study of hyperbolic 3-manifolds.
