Symplectic Geometry

This is a one-semester graduate course taught in Spring Semester 2020 at the East China Normal University and Fall Semester 2021 at Saint Petersburg State University.

Lecture notes available here.

I. Foundations

  1. Hamiltonian Functions
  2. The Structure of Symplectic Manifolds
  3. Group Actions and Hamiltonian Manifolds
  4. Reduction
  5. Reduction by Stages

II. Geometry of the Moment Map

  1. Morse Theory
  2. The Moment Polytope
  3. Delzant's Theorem
  4. Connections and Curvature
  5. The Duistermaat–Heckman Theorem
  6. The Exact Stationary Phase Approximation

III. Geometric Quantization

  1. Prequantization
  2. Complex Manifolds
  3. Kähler Quantization