The common theme of this differential topology project is to study invariants of 3- and 4-dimensional manifolds, their relations to the theory of transformation groups, geometry of submanifolds, quantum physics, moduli spaces of instantons, and related questions on cohomology of discrete groups. The investigator will continue his work with Sylvain Cappell and E. Y. Miller on the theory of generalized Casson flows, on Witten-Reshetikhin-Turaev invariants, on spectral flow invariants of 3-manifolds, and on the application of symplectic geometry techniques in Atiyah-Patodi-Singer index theory. Jointly with Dariusz Wilczynski, he intends to investigate representing homology classes of 4-manifolds by surfaces of minimal genus. A related topic is the joint work of I. Hambleton and the investigator on the theory of smooth transformation groups on 4- manifolds, equivariant moduli spaces of instantons, and the equivariant Donaldson invariant. Together with Steven Weintraub, he plans to continue their work on the cohomology of arithmetic subgroups of Sp4 and also on an invariant of ramified covering spaces originating from the work of J.-P. Serre. Finally, with Alan Brownstein, the investigator plans to study the motion group of n-strings in 3-space, its cohomology, and related moduli spaces. Traditionally, differential topology has provided a means of analyzing certain questions about the world we live in, its geometry and its physics, but in recent years the situation has become much more fluid, with physics often providing useful geometric notions as much as vice versa. The latest such foreign imports so to speak have been the Donaldson invariant of 4- manifolds and related invariants discovered since but along the same lines. There is, of course, a long history 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 these tools 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 a variation on this theme. 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 modifications of 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. Work along these lines will be among the projects undertaken by the investigator with some of his legion of collaborators.
