Notre Dame, Fall 2019

Talk notes

Characteristic classes:

- Bundles and connections.
- Stiefel-Whitney classes and Grassmanians.
- Characteristic classes.
- Group cohomology, BG.
- Chern-Weil homomorphism.
- Equivariant cohomology.

- Symplectic manifolds, Lagrangian submanifolds, Moser's trick.
- Moment maps.
- Convexity theorem (Delzant, Atiyah-Guillemin-Sternberg), moment map polytopes.
- Moduli space of flat connections. Case of a surface, Atiyah-Bott symplectic structure.
- Floer homology.
- Geometric quantization.
- Geometric quantization of the moduli space of flat connections on a surface, Hitchin's connection.

- cohomology operations, Steenrod algebra.
- Einstein equation.
- Gromov-Witten invariants, part I. Part II.

- Dirac operators, Clifford algebras, Spin structures.
- Elliptic operators.
- Cobordism ring, genera, multiplicative sequences.
- Atiyah-Singer index theorem (formulation via char classes).
- Hirzebruch signature theorem and Hirzebruch-Riemann-Roch theorem.
- Topological K-theory, topological index of an elliptic operator.
- Index theorems on manifolds with boundary.

Exercise sheet.