Index theory and Non Commutative
Geometry
Abstract
This course is an introduction to Non-commutative Geometry (NCG)
from the point of view of index theory. I will explain the different versions of the
fundamental Atiyah-Singer Index Theorem and how they can fit in the NCG point of view. I
will also try to show that new index formulaes can be reached by the use of the Connes
point of view.
Index
1. REVIEW OF THE COMMUTATIVE INDEX THEORY
a) The Index Theorem for Toeplitz operators
b) The classical geometric operators
c) The Index Theorem for 4-manifolds
d) The general Atiyah-Singer formula
e) Some commputations and corollaries
f) The Atiyah-Segal Lefschetz formula
g) Other developpments; some conjectures
2. THE NONCOMMUTATIVE APPROACH
a) Von Neumann Fredholm modules
b) The analytic Chern-Connes character
c) Von Neumann spectral triples: examples
d) The diff-equivariant example
3. RESIDUES AND THE CONNES-MOSCOVICI INDEX THEOREM
a) Dixmier trace and locality
b) Homology of complete symbols and residue cocycles
c) The Noncommutative von Neumann Index Theorem
d) Corollaries and examples
Appendix A: Ideals in von Neumann algebras and von Neumann Dixmier trace.
Appendix B: Cyclic cohomology.
Appendix C: A survey of $K$-theory.
Appendix D: A survey of Characteristic Classes.
Lectures
1. Index theory and Non Commutative
Geometry
