**Mohammed Berkani**

*University Mohammed I*

https://videoconf-colibri.zoom.us/j/9301932415

Title: A new approach in index theory

**Abstract:** In this talk, we define an analytical index for a continuous family of Fredholm operators parameterized by a topological space X into a Hilbert space H, as a sequence of integers, avoiding in this way the use of K-theory as in the work of Atiyah-Singer and extending naturally the usual definition of the index. Then we prove the homotopy invariance of the index and we give an Atiyah-Janich type theorem in the case where $X$ is a compact, locally connected space. Extension to the case of Banach spaces will be discussed and several open questions will be pointed out.