This award supports the research of Peter Haskell (joint with Jeffrey Fox of the University of Colorado) on equivariant KK-theory. Equivariant K-theory and the K-theory of crossed product algebras have played an important role in the solution of geometric and topological problems associated with group actions and with fundamental groups of manifolds. Examples of such problems include many cases of the Gromov-Lawson-Rosenberg conjecture on the existence of metrics with positive scalar curvature and many cases of the Novikov conjecture on the homotopy invariance of higher signatures. Kasparov's representation ring KK(G)(C,C) and its distinguished idempotent have played a fundamental role in this program. This research will apply the Kasparov representation theory with the intention for contributing to the geometric understanding of the K-theory of crossed product C(*)-algebras. This work in 'modern analysis' blends sophisticated research in algebra, geometry, and analysis to examine the underlying theory of manifolds or surfaces. This analysis is by way of the action of the transformation groups of a space on a surface or geometric object in that space.
