Math 257B: Symplectic Geometry and Topology
Lectures: Eleny Ionel, Tue and Th 10:30am-11:50am 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 blow-up, 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:
- Atiyah-Bott, "The moment map and equivariant cohomology"; see also the survey "Morse Theory Indomitable" by Bott.
- Duistermaat-Heckman, "On the Variation in the Cohomology of the Symplectic Form of the Reduced Phase Space".
- Gromov, "Pseudo holomorphic curves in symplectic manifolds".
- Robbin-Salamon, "The Maslov index for paths".
- D. Salamon, "Lecture notes on Floer Homology".
Recommended Problems: HW1, HW2,
- 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 blow-up, symplectic (fiber) sum.
- Week 7: Pseudo-holomorphic 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.