### Groupoid C*-algebras and index theory on manifolds with singularities, by Jonathan Rosenberg

The simplest case of a manifold with singularities is a manifold *M*
with boundary, together with an identification of the boundary
with a product ß*M* × *P*, where *P*
is a fixed manifold. The associated singular space is obtained by
collapsing *P* to a point. When *P* =
**Z**/*k* or *S*^{1},
we show how to attach to such a space a
noncommutative C*-algebra that captures the extra structure. We
then use this C*-algebra to give a new proof of the Freed-Melrose
**Z**/*k*-index theorem and a proof of an index theorem for manifolds
with *S*^{1} singularities. Our proofs apply to the real as
well as to the complex case. Applications are given to the study of metrics of
positive scalar curvature.

Jonathan Rosenberg <jmr@math.umd.edu>