Math 257A: Introduction to Symplectic Topology and Geometry

Autumn 2010

Lectures: Eleny Ionel, Tue and Th 9:30am-10:45am, room Herrin T195;

Course Description: This course is an introduction to symplectic and contact topology and geometry, part of a two quarter sequence. The topics we plan to cover this quarter include:
- Symplectic manifolds, symplectic and Lagrangian submanifolds, symplectomorphisms: examples, obstructions and properties; Gromov nonsqueezing theorem;
- Hamiltonian diffeomorphisms and Hamiltonian flows; fixed points and periodic orbits, Lagrangians intersections and the Arnold conjectures;
- Almost complex structures, symplectic vector bundles, the first Chern class and the Maslov index;
- Constructions of symplectic manifolds: symplectic fibrations, Lefshetz fibrations, symplectic blow-up, symplectic (fiber) sum;
- Contact manifolds, Legendrian submanifolds, Reeb flow and the Weinstein conjecture.
- Symplectic and Hamiltonian group actions, moment maps, symplectic quotients and fixed points of Hamiltonian group actions.
Recommended reading: We will not follow any references too closely but here are some suggestions:
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, (second edition), Oxford, 1998.
- A. Cannas da Silva, Lectures on Symplectic Geometry, Springer (2001).
Lecture notes available on the web:
- A. Cannas da Silva, overview written for the Handbook of Differential Geometry, available from her website
- Kai Cieliebak, lecture notes for Math 257A (winter 2001), available from his website
Further references:
- V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition , Springer-Verlag, New York (1989).
Recommended problems: PS1, PS2, PS3

Here is the link to Alex's lecture notes;