Speaker
Description
In numerous PDE problems, differential operators naturally appear with some geometrical constraints. For instance, if $(M, \mathcal{F})$ is a foliated manifold one may be interested in operators acting longitudinally on the leaves, if $M$ has borders (or corners) one may ask operators to be well-behaved around the borders, on a manifold endowed with an action of a Lie group $G$ one may consider $G$-equivariant operators...
The aim of this talk is to explain how Lie groupoids provide a unified framework to study the analytic behaviour of all of these operators. This point of view allows to adapt the notions of principal symbols and ellipticity to these contexts and to define a well behaved pseudodifferential calculus in order to build parametrix. One can then use these operators to build classes in the K-theory of the $C^*$ algebras of the groupoids, as a first step towards Atiyah-Singer type theorems.