Classical theorem of Atiyah-Singer expresses the index of the elliptic operator on a compact manifold M through some topological invariants of M. We will prove a particular case of this theorem - the Gauss-Bonnet-Chern theorem. In our proof we will use ideas of supersymmetry, which came into mathematics from physics. In the second part of our course (time permitting) we will outline the proof of the Atiyah-Singer theorem for Dirac operators by means of functional integration.
All necessary facts from differential geometry and analysis will be discussed during lectures. Some knowledge of differential forms and de Rham cohomology will be useful.
1. Short course in Riemannian Geometry, including parallel translation and curvature operators.
2. Notion of supersymmetry, Berezin-Patodi formula.
3. Bochner Laplacian and Weitzenbock formula.
4. Elliptic regularity (description of results).
5. Localization of the heat kernel and the completion of the proof of the Gauss-Bonnet -Chern theorem.
6* Path integrals.
7* Atiyah-Singer formula for Dirac operator.
