Abstract Guentner The proposed research comprises three distinct projects; coarse geometry and index theory of elliptic operators on homogeneous spaces, topological invariants of quantum mechanical systems, and the construction of an equivariant version of the E-theory groups defined by A. Connes and N. Higson. All three involve the notion of asymptotic homomorphisms of C*-algebras which provide an elegant realization of K-homology, the generalized homology theory dual to K-theory. The first is motivated by work of P. Baum, R. Douglas and M. Taylor calculating the image of the K-homology class of a first order, elliptic differential operator on a manifold with boundary under the boundary map in K-homology. The second is based on the observation that a quantization of a classical mechanical system based on a continuous parameter of Planck's constant may be used to construct an asymptotic morphism and hence an element of a K-homology group. The third is a joint project with N. Higson and J. Trout. Our equivariant groups will prove useful in a number of contexts. In particular they have applications to the Baum-Connes conjecture. The proposed research deals with index theory on non-compact, complex manifolds possessing a large number of symmetries. The geometric structure of the boundary of these manifolds plays an important role in index theory. I propose to study its precise relationship to the analytic properties of elliptic differential operators on the manifolds themselves using the techniques of coarse geometry and the newly developed E-theory. The second project is based on the close relationship between elliptic differential operators on these manifolds to certain quantum mechanical systems. My methods can be used to obtain new topological invariants of these systems and relate them to index theory. The research should lead to a better understanding of a number of quantum mechanical systems from mathematical physics and also has bearing on a number of outstanding problems in index theory including the Baum-Connes Conjectupe.
