Room P3.10, Mathematics Building

Catarina Carvalho, Instituto Superior Técnico
Operators on groupoids and the analytic index

In this talk I will discuss a generalization of the Fredholm index for operators on groupoids. A (pseudo)differential operator on a Lie groupoid is a family of (pseudo)differential operators on $s$-fibers, satisfying invariance and smoothness conditions. The analytic index of such an operator is defined as a $K$-theory map, using the normal groupoid construction and $K$-theory for $C$*-algebras. For the pair groupoid, one recovers operators on manifolds and the Fredholm index. The main non-trivial examples to keep in mind are Melrose's $b$-groupoid approach to the index theorem on manifolds with boundary, and Connes and Skandalis index theorem for operators on foliated manifolds.