In the first project, "Torsion Invariants and Algebraic K- theory of von Neumann Algebras", a new topological invariant is defined for a Riemannian manifold with a proper cocompact action ofa countable discrete group G by isometries. It generalizes classical notions of Reidemeister torsion. One goal is to find a better understanding of the weak K-theory of the von Neumann algebra of G, where the invariant lives. The principal investigator wants to show by computation that the invariant carries a lot of information in interesting cases, including homology 3-spheres, hyperbolic manifolds, crystallographic manifolds and symmetric spaces. Another problem is to give an analytic interpretation in terms of the spectral theory of the Laplace operator. The other project, "Surgery Transfer and a General Signature Formula for Fibre Bundles," is devoted to the study of the surgery transfer of a fibre bundle of manifolds. An algebraic description was developed by the principal investigator and Andrew Ranicki. It will be used to make explicit calculations, to prove vanishing results, which have geometric meaning and applications, and to establish a general signature formula for fibre bundles. The first project is a joint project together with Professor Melvin Rothenberg of the University of Chicago, the second with Professor Andrew Ranicki of the University of Edinburgh. Both combine topology, analysis, and algebra in genuine synthesis. Their principal value may be generating new tools for understanding manifolds and their symmetries. Manifolds are, of course, very basic geometric objects and arise almost everywhere one looks in mathematics and physics, often as solutions of systems of ordinary or differential equations.
