Seminar

The Atiyah-Singer index theorem expresses the index of the three fundamental geometric operators,

i.e. the signature operator, the spin-Dirac operator and the Riemann-Roch operator, in terms of

the L-genus, the A-roof genus and the Todd genus of the manifold. These genera have remarkable stability properties.

One can define higher genera by taking the fundamental group and its cohomology into account and deep results of Kasparov, Connes, Moscovici and many others express these numbers in terms of the K-homology class [D] associated to these operators, information that is then crucial in studying the stability properties of the higher genera.

In this talk I will begin by explaining this fascinating circle of ideas, together with its geometric applications, and then move on and give an answer to the following question: what of all this can be generalized to singular manifolds? As we shall see the answer depends heavily on microlocal analysis, at least for the first 2 examples.

The results that I will present are scattered in papers with Albin-Leichtnam-Mazzeo, Botvinnik- Rosenberg, Bei.