R.L. Cohen plans to pursue several projects involving algebraic topological questions arising in gauge theory. One project, joint work with J.D.S. Jones, involves finding an explicit relationship between representations of braid groups and families of elliptic operators coupled to SU(2) monopoles. Another project is to pursue joint work with Jones and G. Segal on Morse theory. Given a Morse function on a compact manifold, they have described an explicit simplicial space decomposition of the manifold using classifying space theory. They plan to investigate the applicability and implications of these techniques to various infinite dimensional settings; for example, the Chern-Simons functional and the Yang-Mills functional. Milgram plans to study the geometry of moduli spaces of holomorphic bundles, concentrating on SU(2)-instantons over S4 regarded as equivalence classes of based holomorphic bundles on CP1 x CP1. He also expects to continue his study of the geometry and homology of sporadic groups and exceptional Lie groups. Finally, he hopes to work with R. Penner on the relations between moduli spaces of Riemann surfaces and representations of symmetric groups. Mrowka plans to pursue three projects. The first is to prove there are 4-manifolds homotopic to a K3 surface which do not admit any complex structures. The second is to study the effect on the Donaldson polynomial of the existence of embedded S2's or T2's. Finally, with K. Walker, he hopes to study gauge- theoretic generalizations of the Casson invariant. The connections between geometry and physics involved in this work are striking. They work both ways. While much of the motivation for studying four-dimensional manifolds, Lie groups, and group representations may once have come from the hope of calculating things of interest to physicists, in recent years notions inspired by quantum physics have given rise repeatedly to constructions of new topological invariants that answered questions the topologists could not answer before.
