This group project in topology deals with research in a wide range of topics in topology and related areas of algebra, algebraic K-theory, and geometry. May studies various areas centering on global structures in stable homotopy theory and related fields. He and his collaborators have opened up stable homotopy theory to serious point-set level algebraic study with their recent new approach to highly structured ring and module theory. Rings, modules, and algebras can now be defined as objects with well-behaved products in a symmetric monoidal category of spectra, allowing many constructions and applications that were not possible previously. He and Greenlees have opened up new interactions between equivariant and non-equivariant stable homotopy theory with their analysis of Tate cohomology and their proof of a completion theorem for module spectra over MU. Rothenberg studies analytic and combinatorial torsion invariants in geometric topology. He and his collaborators have explored generalizations of the classical invariants to equivariant and non-compact situations and have constructed torsion invariants for fiber bundles with compact fibers, these being cohomology classes rather than just numerical invariants. Weinberger's work centers on geometry and analysis on compact spaces with singularities and on noncompact manifolds. His surgery theory on stratified spaces can be applied directly to various problems and has led to a changed perspective on group actions, both in terms of answering old questions and in formulating new ones. The perspective is completely integrated with the theory of homology manifolds at the level of conjecture, and somewhat at the level of theorem. Furthermore, detailed analysis of what would be involved in proving such conjectures seems to have deep connections with logic, complexity theory, and non-commutative geometry. A major emphasis of the project is graduate education. With three junior faculty and fifteen current graduate students in topology at Chicago, the topology program supported by this grant is one of the world's largest. In the three years 1996-98, it has graduated ten new PhD's, with 1998-99 jobs at MIT (2), Michigan (2), Illinois (2), Berkeley, CUNY, Rutgers, and Utah. Students supported on the grant work in a wide variety of areas of algebraic and geometric topology, and some of them work on the interfaces between algebraic topology and algebraic geometry on the one hand and between geometric topology and differential geometry on the other. Although focused on topology, the work supported by this grant impinges on many other areas of mathematics. Some of it also impinges on current work in mathematical physics, where the kinds of topological and geometric structures studied by the investigators have direct relevance. ***
