Topics in low-dimensional topology

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Abstract Award: DMS-0104039 Principal Investigator: Darren Long The proposers plan to continue their work on a variety of problems bearing on the understanding of the geometric and topological aspects of low dimensional manifolds. On the topological side, the areas of study include Heegaard splittings, unknotting tunnels and related topics; the geometric side deals with issues related to orbifolds, the construction and melding of surface subgroups, developing an understanding of higher dimensional hyperbolic manifolds, as well as some interactions with other areas, for example algebraic number theory. Manifolds play a central role in physics, mathematics and to some extent in other sciences, since they are objects which on small scales look like Euclidean space of dimension n. For example, the space that we live in is a three manifold and space-time is a four manifold. For this and other reasons, these dimensions have attracted a good deal of attention in mathematics and physics. One basic unsolved problem is exactly which manifold is the correct model for the universe - there has been some speculation that it falls into the class of so-called hyperbolic manifolds, a certain class of three dimensional spaces which in some sense appear to be generic. Cosmological theories about the origins of the universe put constraints on the shape of space-time and hence constraints on which manifolds could occur. One of the goals of this project is refine current methods (which have already ruled out many possibilities) to narrow the search down further. There are also other applications of geometric ideas which are less obvious but still, in fact, directly important. It is no exaggeration to say that almost any problem which can be formulated qualitatively can be studied with geometric methods and as a result the powerful tools developed over the last thirty years in low-dimensional topology can be brought to bear upon it.