Next lecture: 11:15 a.m., Friday October 21, 2011. "The Dirac operator and the Euler characteristic of a compact manifold"

- Coordinates: Lecture Hall A21, Department of Mathematics, UoA, Panepistimiopolis.
- Aim: to present Alain Connes' work on the analytic index of elliptic pseudodiffernetial operators on a (regular) foliation. Roughly, what is the analytic index, what information it carries about the nature of the foliation, and how to compute it (Atiyah-Singer index theorem).
- Winter semster: In fact, we won't touch foliations until the spring semester. It will take the entire winter semester to understand the case of manifolds, introducing various notions and giving the overall feeling of the theory in that easy case. A tentative list of topics to be covered in the winter semester is:

- Fredholm operators
- Pseudo-differential operators (definition, ellipticity, analytical index)
- K-theory of C*-algebras (definition and relation with topological K-theory, exact sequences, Bott periodicity)
- Topological index (definition, Chern character, characteristic classes)
- Equality of the two indices (Atiyah-Singer theorem)

- Spring semester: During the spring semester we'll go through Connes' work on (regular) foliations. A very rough skeleton of the seminar for that period is:

- Introduction to (regular) foliations (definition, examples, the notion of holonomy, the holonomy groupoid and its C*-algebra)
- Transverse measures (holonomy invariant measures, Ruelle-Sullivan current)
- Locally traceable operators
- Connes' measured index theorem for foliations.