Math 369: Index Theory

The course will be a tour through the mathematics surrounding index theory. We will discuss characteristic classes, the Chern-Weil homomorphism, pseudodifferential and elliptic operators on compact manifolds. We will prove the local analytic version of the Atiyah-Singer index theorem and discuss applications of the index theorem to differential and algebraic geometry and to topology. Depending on time and interest we may discuss anomalies and/or spectral invariants such as the eta invariant. I will assume a basic course in differential geometry and algebraic topology. I will review elliptic pde to the extent needed by the class.