The course will contain the operator K-theory approach to the Atiyah-Singer index theorem. We will start with some classical examples of elliptic differential operators on compact smooth manifolds without boundary. This will naturally lead us to two special cases of KK-theory: K-cohomology and K-homology.
KK-theory will be introduced gradually, as much as it is needed for index theory. Most examples will come from differential and pseudo-differential operators. Large part of technical results related with $KK$-theory will be given without proof: because time is limited, and also because we need $KK$-theory for this course only as a tool.
Other technical tools include Clifford algebras and Dirac operators. Although all definitions will be given in the course, I advise the listeners to consult the book ``Spin geometry'' by H. B. Lawson and M.-L. Michelsohn on these issues.
The main part of the course will contain a proof of the $K$-theoretic version of the Atiyah-Singer index theorem. The cohomological Atiyah-Singer index formula for compact manifolds will be obtained as a corollary. Various applications will be discussed as much as time allows.
1. M. F. Atiyah, I. M. Singer: ``The index of elliptic operators'', I, III, Annals of Math., 87 (1968), 484-530, 546-604.
2. H. B. Lawson, M.-L. Michelsohn: ``Spin geometry'', Princeton Univ. Press, 1989.