In this course we will cover the basics of the theory of elliptic operators, examples of which include the de Rham and Dolbeault operators, as well as their generalisations such as the Dirac operator. We will explain how several seemingly unrelated results in geometry and topology (e.g. the Hirzebruch and Rokhlin signature theorems and the Riemann-Roch theorem) all follow from the general index formula by M. Atiyah and I. Singer, and sketch a proof of the latter.

^*This course is to be read in the Spring 2017 semester only.

Prerequisites: Smooth manifolds and singular cohomology.

Curriculum:

1. Vector bundles and characteristic classes: a summary of results.
2. Differential operators: the definitions and first examples.
3. Elliptic operators: the definition and basic properties.
4. Riemannian metrics on manifolds and the de Rham operator.
5. The signature operator.
6. Complex manifolds and the Dolbeault operator.
7. Clifford algebras and their representations.
8. Reduction of the structure group of a vector bundle. Spin structures on vector bundles.
9. The Dirac operator.
10. Elliptic regularity and related results about elliptic operators on compact manifolds.
11. First applications: the de Rham and Hodge decomposition theorems; Serre duality.
12. The Atiyah-Singer index formula.
13. Applications of the index formula: the Riemann-Roch theorem, the Hirzebruch signature theorem, Rochlin's signature theorem.

Textbooks

• Algebraic Topology by A. Hatcher, freely available online at http://www.math.cornell.edu/~hatcher/AT/ ATpage.html.
• Spin geometry by H.B. Lawson and M.-L. Mischelsohn.
• Seminar on the Atiyah-Singer index theorem by R.S. Palais et al.
• Characteristic classes by J. Milnor and J. Stashe.
• The Atiyah-Singer index theorem by P. Shanahan.
• Differential Analysis on complex manifolds by R.O. Wells.