Abstract Trout The proposed research involves defining an equivariant ``Atiyah-Singer''-type topological index for elliptic differential operators which are invariant with respect to the proper and cocompact action of a locally compact group on the underlying manifold, i.e. this index arises from equivariantly embedding the base manifold into a (possibly infinite dimensional) Euclidean representation of the group and applying an appropriate version of equivariant Bott periodicity. The proof of the corresponding equivariant index theorem, namely that this topological index is equal to G. Kasparov's equivariant analytic index for the elliptic operator, requires the use of equivariant asymptotic homomorphisms of C*-algebras and the associated equivariant version of the E-theory groups of A. Connes and N. Higson. This research has applications to the Baum-Connes Conjecture and the Novikov Conjecture. The proposed research deals with the index theory of elliptic differential operators on manifolds which commute with the action of a locally compact transformation group, i.e., studying the spaces of solutions of differential equations which exhibit certain degrees of internal symmetry. Associated with each elliptic operator is an "analytic index," which can carry important topological and geometric information about the manifold the operator is associated with. However, this analytic index is very difficult to compute in general, requiring a great deal of detailed analysis. I propose to construct, in a more topological and geometric fashion, another method for computing this index, generalizing the case of compact groups considered by Atiyah and Singer. Being able to compute the indices of these elliptic operators should lead to a clearer understanding of a variety of problems in geometry and representation theory, which are important to physics. In particular the project will have a bearing on the Baum-Connes Conjecture and the still-unsolved Novikov Conjecture.
