Lecture | Thursdays | SG 3-12 | 15:15 - 16:45 |

Seminar | Tuesdays | SG 3-10 | 17:15 - 18:45 |

Please register for the course with a short e-mail to boldt ät math.uni-leipzig.de.

Course description: Spin geometry is a special topic within Riemannian geometry at whose core lies the Dirac operator. The Dirac operator was found by P. A. M. Dirac while he was in search of a physical theory of the electron which encompasses both special relativity and quantum mechanics. It took mathematicians several decades, namely until after the establishment of the framework of principal fibre bundles, to realize how the Dirac operator can be defined on a Riemannian manifold. The driving force behind this quest was M. Atiyah's and I. Singer's desire to answer a question of Gelfand: can the analytic index of a differential operator be expressed in topological quantities of the underlying compact manifold. Their answer resulted in the Atiyah-Singer-Index Theorem, one of the greatest achievements of 20th century mathematics.

The course aims at introducing the students to the fundamentals of Riemannian spin geometry so that they can navigate the literature and easily connect to current research. Prerequisite is an understanding of the basics of Riemannian geometry as taught in Differential Geometry I.

Please note: If you are part of the Mathematical Physics Master's program you also have to attend the lecture Morse theory, closed geodesics and geometry by Prof. Dr. Hans-Bert Rademacher to successfully complete the module Advanced Differential Geometry 2.

Format of the lecture: Currently, the plan is to divide the students into groups of at most eight, one of which will be physically present in the lucture room with the rest of the audience attending the lectures online. The groups will be cyclically permuted on a weekly basis.

Exercise sheets: There will be weekly exercise sheets whose solution will be discussed in the seminar on Tuesdays.

Exercise sheet no. 1

Exercise sheet no. 2

Exercise sheet no. 3 (corrected version)

Exercise sheet no. 4

Exercise sheet no. 5

Exercise sheet no. 6

Exercise sheet no. 7 (corrected)

Exercise sheet no. 8

Exercise sheet no. 9

Exercise sheet no. 10

Lecture notes: 2021/02/04

Bibliography:

H. Blaine Lawson, Jr., Marie-Louise Michelsohn: Spin Geometry. Princeton University Press, Princeton, NJ, 1989.

Thomas Friedrich: Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics, 25. American Mathematical Society, Providence, RI, 2000.

Nicolas Ginoux: The Dirac spectrum. Lecture Notes in Mathematics, 1976. Springer-Verlag, Berlin, 2009.