Differential geometry studies smooth manifolds with additional structure, and using additional structure. The basic ingredients in this are the (co)tangent bundle and differential forms, and differential operators like the de Rham operator, the Hodge-star operator and the Laplacian.
A next step, with a fascinating blend of algebra, analysis, geometry and topology is the introduction of spin structures.
Spin structures are a refinement of orientiations. They are based on the use of the Lie group Spin(n) (instead of SO(n)) and their principal bundles. They can be constructed and understood via the Clifford algebras, or topologically via covering theory. Using basics of the representation theory of Clifford algebras, one can construct a fundamental vector bundle (the spinor bundle) on each manifold with a spin structure. Even better, there is also a fundamental differential operator, the Dirac operator. Indeed, it is the global analogue of the square root of the Laplacian constructed by Dirac.
This operator has very nice analytic properties: it is an elliptic operator, in particular its spectrum is very controlled.
In the seminar, we will introduce and study in depth the different algebraic/geometric ingredients mentioned above, then develop the Dirac operator and its analytic properties. The seminar will cumulate in the celebrated Atiyah-Singer index theorem for the Dirac operator --for time reasons without an in-depth treatment of all analytic details.
Seminar program
| Title | Reference | math. area | ||
| (real) Clifford algebras and Spin: basic properties I | I.3, I.2, (I.4) | algebra | ||
| (real) Clifford algebras and Spin: basic properties II | I.3, I.2, (I.4) | algebra | ||
| representations of Cl(n) and Spin | I.6, I.7.1-I.7.2 | algebra | ||
| vector bundles, short intro to K-theory | I.9 | algebra+topology | ||
| spin structures on bundles and Stiefel-Whitney classes | II.1 | topology | ||
| spin manifolds (no spin bordism) | II.2 | topology | ||
| Clifford and spinor bundles | II.3 | topology, algebra | ||
| connections on spinor bundles, curvature I | II.4 | differential geometry | ||
| connections on spinor bundles, curvature II | II.4 | differential geoemtry | ||
| Dirac (type) operators | II.5 (up to 5.7) | differential geometry, some analysis | ||
| The fundamental Dirac type operators | II.6, part of II.5 | differential geometry, some analysis | ||
| Sobolev spaces and (pseudo)differential operators | III.1-III.3 | analysis | ||
| elliptic operators and their index | III.4-III.6 | analysis | ||
| Atiyah-Singer index theorem | III.11, III.13 | analysis, topology |
Literature