250119 VO Spinors and Dirac Operators (2023W)

Lecture course on the basics of Spin geometry and Dirac operators. We will start with the case of low dimensions, where the spin groups can be described in terms of classical Lie groups and basic Dirac operators can be constructed by hand. Then we will move to general dimensions, discussing the general concept of spin structures and Dirac operators and their relation to Riemannian geometry. The question of existence and uniqueness of Spin structures gives rise to connections to algebraic topology and characteristic classes. Finally we will discuss the general construction of Spin groups via Clifford algebras and hint on the role of Dirac operators in index theory.

The main background required for the course is analysis on manifolds and the basics of differential geometry and Lie group theory. Background on Riemannian geometry, deeper aspects of Lie theory and algebraic topology can be helpful but is not required.