Cornell Math - Math 653 (SP01)

Math 653 — Spring 2001
Differentiable Manifolds: the Atiyah-Singer Index Theorem

The Atiyah-Singer Index Theorem is a landmark of twentieth-century mathematics. It reveals deep links between geometry (connections and curvature), topology (K-theory and characteristic classes), and analysis (linear partial differential equations), and plays an important role in all three subjects.

The goal of the course is to understand the statement of the theorem and explore some of its applications, such as the Gauss-Bonnet Theorem, the Signature Theorem and the Hirzebruch-Riemann-Roch Theorem. We shall also examine the ideas behind the proof, emphasizing the geometric and topological aspects.

One could easily spend a whole semester developing the basic theory needed to state this imposing result and end up not even understanding what it is all about. The only way to avoid this is to abandon all hope of giving an exhaustive step-by-step treatment. I shall therefore at times have to quote facts from related areas, such as Fredholm theory and K-theory, and ask the audience to accept them on faith.

Those who want to take the course for credit will be asked to give one or two lectures towards the end of the semester. Suitable topics include applications of the Index Theorem, and any of the results quoted without proof in class.

Prerequisites: familiarity with manifolds, vector bundles and differential forms. Knowing the de Rham Theorem and the Hodge Theorem would be helpful.