**Partha Chakrabarty**

Institute of Mathematical Sciences, Chennai

April 21, 2011

**Around the local index formula**:
Connes-Moscovici Local Index Formula expresses the index pairing in terms of sums of certain residues. However this formula can be applied if certain hypothesis holds. I want to talk about a weakening of the heat kernel expansion and a transformation which implies stability of the hypothesis. I will definitely talk about the basic ingredients of the subject and a quick application by Connes, of some of the ideas towards the no idempotent conjecture for the group ring of free group on two generators. After that if time permits I'll talk about the local index formula.