Index Theory of Pseudodifferential Operators Onlie Structures

We review recent progress regarding the index theory of operators defined on non-compact manifolds that can be modeled by Lie groupoids. The structure of a particular type of almost regular foliation is recalled and the construction of the corresponding accompanying holonomy Lie groupoid. Using deformation groupoids, K-theoretical invariants can be defined and compared. We summarize how questions in index theory are addressed via the geometrization made possible by the use of deformation groupoids. The discussion is motivated by examples and applications to degenerate PDE’s, diffusion processes, evolution equations and geometry.