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.