MATH 70330, Intermediate Geometry and Topology
Notre Dame, Fall 2019

Talk notes

Characteristic classes:

1. Bundles and connections.
2. Stiefel-Whitney classes and Grassmanians.
3. Characteristic classes.
4. Group cohomology, BG.
5. Chern-Weil homomorphism.
6. Equivariant cohomology.

Symplectic geometry:

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

Extra talks:

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

Index theory

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

Exercise sheet.