Lecturer: Andreas Hermann

The goal of this course is to give a proof of the Atiyah-Singer index theorem, which is regarded as one of the main results of 20th century mathematics. We start by introducing Clifford algebras, spin groups and their representations. Then we prove important results on the analysis of Dirac and Laplace type operators. After that, we study the fundamental solution of the heat equation, which will play an important role in the proof of the index theorem.

Place and time:
Tuesday 08:15-09:45 in 2.09.0.14
Friday 08:15-09:45 in 2.09.1.10

Exercise class:
Thursday 12:15-13:45 in 2.09.1.10 (Max Lewandowski)

Exercise sheets:
Moodle-Link

Semester (recommended):
ab 7.

Modulnummer(n):
81j, 771, 772, 781, MATVMD815, MATVMD611, MATVMD612

Prerequisites:
Some familiarity with basic notions of differential geometry (e.g. manifolds, vector bundles) will be required.

Literature:
Detailed lecture notes will be provided. Furthermore the following books are recommended:
1. C. Bär: Differentialgeometrie, Vorlesungsskript, Potsdam 2006
2. N. Berline, E. Getzler, M. Vergne: Heat kernels and Dirac operators. Springer 2004
3. J. Roe: Elliptic operators, topology and asymptotic methods. Second edition. Longman 1998
4. H. B. Lawson, M.-L. Michelsohn: Spin geometry. Princeton University Press 1989
5. T. Friedrich: Dirac operators in Riemannian geometry. AMS 2000