Ben Webster: Symplectic Geometry

This is a 24-lecture course, with each lecture being about 90 minutes or so, given online by Ben Webster.

This class covers the basic theory of symplectic manifolds. Symplectic structures play a key role in modern mathematics and physics. We will discuss their basic local theory (in particular, the Darboux theorem), connections to complex and Kähler geometry, Hamiltonian mechanics, moment maps and symplectic reduction, and some additional topics, such as toric varieties, hyperkähler structures, quantization, Fukaya categories and mirror symmetry.

Prerequisites: Familiarity with the basics of differential geometry: smooth manifolds, tangent vectors and forms. In particular, exterior and Lie derivatives will play an important role. Some knowledge of Lie groups and Lie algebras will also help, though we will briefly discuss the required background.

1. Introduction to the course
2. Symplectic forms on cotangent bundles/Lagrangian submanifolds
3. Moser theory
4. Darboux-Weinstein theory
5. Almost complex manifolds
6. Complex manifolds
7. Kähler manifolds
8. Hamiltonian vector fields
9. Hamiltonian and Lagrangian mechanics
10. Review of Lie theory and actions
11. Moment maps
12. Poisson geometry
13. Symplectic reduction
14. The moment map in gauge theory and Yang-Mills
15. Toric manifolds/varieties 1
16. Toric manifolds/varieties 2
17. Hyperkähler manifolds
18. Hyperkähler quotients
19. Deformation quantization
20. Geometric quantization
21. Geometric quantization/Mirror symmetry
22. Mirror symmetry for tori
23. Mirror symmetry 1
24. Mirror symmetry 2

These videos were produced by the Fields Institute and form the course Symplectic Geometry.