Index theory over groupoids, inertia spaces and the cyclic homology theory of convolution algebras

Index theory over singular spaces is an active research area within global analysis and has many potential applications. For spaces which can be described by groupoids, the well-known index theorems for orbifolds indicate that the so-called inertia spaces encode the contribution of singularities to analytic indices. Moreover, the inertia space of a Lie groupoid encodes interesting topological, geometric, and analytic information about the original Lie groupoid. It is the goal of the talk to explain this point of view using non-commutative geometry as a unifying tool. In particular a Hochschild-Kostant-Rosenberg type theorem for the Hochschild homology of the convolution algebra of a proper Lie groupoid is indicated. The talk is based upon joint work, partially in progress, with H. Posthuma and X. Tang, as well as with C. Farsi and Ch. Seaton.