Math 276, Topics in Topology: Lectures on the

Atiyah-Singer Index Theorem, Spring 2011

 

Instructor: Peter Teichner

Lectures: TuTh 12:30-2:00, 51 Evans

Office Hours: Mo 3:00-4:00, 703 Evans

Course Control Number: 54557

Prerequisites: 214, 215A


Syllabus: One could argue that the Index Theorem should be considered to be the best theorem of the previous century: For any elliptic operator on a compact manifold, the analytic index equals the topological index. Roughly speaking, this says that a global analytic datum of a manifold, the number of solutions of a certain linear PDE, can be computed by integrating a local topological expression, a characteristic class, over the manifold.

Motivating examples were the Riemann-Roch theorem for complex curves or more generally, the Hirzebruch-Riemann-Roch theorem for complex manifolds. Similarly, the Gauß-Bonnet theorem for Riemannian surfaces (and its generalization to higher dimensions, the Chern-Gauß-Bonnet theorem) follow from the index theorem.


In this course, we will first discuss those special cases and then go into explaining general elliptic operators and their indices. The index theorem has various generalizations, for example to local, equivariant, respectively family versions.


Class notes written by Dmitri Pavlov. There will be projects introduced in class that will be presented by students at a Mini-Conference on May 16th, 10-6, in 740 Evans.


References include:

Atiyah, Singer, Segal: The index of elliptic operators I-V, Annals of Math. 1968-1971

Hirzebruch: Topological methods in algebraic geometry

Roe: Elliptic operators, topology, and asymptotic methods

Berline, Getzler, Vergne: Heat kernels and Dirac operators

Lawson, Michelsohn: Spin geometry

Palais: Seminar on the Atiyah-Singer Index Theorem