Lectures: Eleny Ionel, Tue and Th 10:30am11:50am in room 381T
Course Description: This course is an introduction to symplectic geometry and topology. We will cover fairly standard material followed by a brief introduction to more advanced topics (holomorphic curves techniques). The main topics we plan to cover in this course include:
 Symplectic manifolds, symplectomorphisms; Lagrangian submanifolds.
 Moser theorem, Darboux theorem, Lagrangian tubular neighborhoods theorem.
 Hamiltonian diffeomorphisms, flux, Hamiltonian flows, isotopy.
 (Almost) Complex Structures, symplectic and complex vector bundles, Chern classes, Maslov index
 Hamiltonian group actions, moment maps and symplectic reductions
 Constructions of symplectic manifolds: symplectic fibrations, Lefschetz fibrations, symplectic blowup, symplectic sums
 Pseudoholomorphic curves and Floer homology (a brief introduction)
This is meant to be mostly an introductory course to the subject, and there are no formal prerequisites for this class, although a good background in topology, geometry and analysis/PDEs is desirable.
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).
Further Readings/Classical papers:
 AtiyahBott, "The moment map and equivariant cohomology"; see also the survey "Morse Theory Indomitable" by Bott.
 DuistermaatHeckman, "On the Variation in the Cohomology of the Symplectic Form of the Reduced Phase Space".
 Gromov, "Pseudo holomorphic curves in symplectic manifolds".
 RobbinSalamon, "The Maslov index for paths".
 D. Salamon, "Lecture notes on Floer Homology".
Recommended Problems: HW1, HW2,
HW3, HW4.
Tentative Schedule:
 Week 1: Overview; Symplectic manifolds, symplectic and Lagrangian submanifolds, symplectomorphisms: examples, obstructions and properties.
 Week 2: Moser's and Darboux theorem, tubular neighborhoods of symplectic/Lagrangian submanifolds.
 Week 3: Hamiltonian diffeomorphisms, flux, Hamiltonian flows, isotopy;
 Week 4: Symplectic and Hamiltonian group actions, moment maps, symplectic reduction.
 Week 5: (Almost) Complex Structures, symplectic vector bundles, Chern classes, Maslov index.
 Week 6: Constructions of symplectic manifolds: symplectic fibrations, Lefschetz fibrations, symplectic blowup, symplectic (fiber) sum.
 Week 7: Pseudoholomorphic curves, Gromov nonsqueezing Thm, sketch of proof.
 Week 8: Local behavior, linearizations; moduli space, transversality.
 Week 9: Bubbling, Gromov compactness, GW invariants
 Week 10: Discussion of the Weinstein conjecture, Arnold conjecture, Lagrangian intersections and Floer theory.