Index theory for (regular) foliationsThis
is the page of my weekly seminar in the Athens math dept. for
2011-12. Announcements about this seminar will appear in this page -- lecture notes (in greek) will be posted here.
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,
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.