The investigator will continue his research on the quantum invariants of 3-manifolds, first defined by Witten (in the context of quantum field theory) and by Reshetikhin and Turaev (in the context of quantum groups). The overall objectives of the project are: 1) to study the structure of the known quantum invariants and their refinements, and to explore possible generalizations; 2) to study connections between quantum invariants and other 3- manifold invariants; 3) to explore the relationship of quantum invariants with number theory; 4) to apply quantum invariants to problems in low dimensional topology. There is a long tradition of mathematicians involving themselves in the problems of physics, often to the mutual enrichment of both subjects. Tools are developed to solve physical problems and then turn out to have much greater generality and become widely used in mathematics. The story of the new quantum invariants of 3-manifolds is similar. Mathematicians interesting themselves in problems of quantum field theory were led to a certain geometric construction. The construction led in turn to an invariant that depended only on the topological character and not the full geometric character of the underlying manifold. There are now variations on the original construction and numerous resulting quantum invariants. Their origin is sufficiently different from that of other previously known invariants that they can be expected to detect things the traditional invariants cannot. It now behooves topologists to demonstrate this as well as to sort out the different quantum invariants and their relationships to more traditional invariants.
