MT 831: Symplectic manifolds

Lecture notes for a graduate course in Fall 2024.

Lecture 1: Overview.

Lecture 2: Overview (part 2).

Lecture 3: Physical movation in R^2n.

Lecture 4: Introduction to symplectic manifolds.

Polterovich and Scherbak wrote a nice recollection about Arnold, the grandparent of symplectic geometry, and the second chapter provides a nice overview of some of the topics we are exploring.

Lecture 5: Darboux's theorem.

Lecture 6: The cotangent bundle.

Lecture 7: The cotangent bundle (part two) and a word on C^n.

Lecture 8: Almost-complex structures and Weinstein's theorem.

Lecture 9: Weinstein's theorem (part two), etc.

Lecture 10: Some of Arnold's conjectures.

Lecture 11: Exact Lagrangians, Hamiltonians, and action.

Lecture 12: Action and some linear algebra.

Lecture 13: Symplectic linear algebra and almost-complex structures.

Lecture 14: Almost complex structures: existence and contractibility.

Lecture 15: Pseudoholomorphic curve basics.

Lecture 16: Symplectic reduction and CP^n.

Lecture 17: Pseudoholomorphic curve basics (part two).

Lecture 18: Energy, area, and the symplectic form.

Lecture 19: Energy, area, and the symplectic form (part two).

Lecture 20: Conformal invariance of energy and the area class.

Lecture 21: The area class (part two).