D. Auroux - Tue. & Thu., 3:30-5pm, Room 35 Evans

**Office:** 817 Evans.

**Office hours:** Mondays 4-6, Tuesdays 5-6 (to be confirmed).

**Homework 1**due Thursday 9/16.**Homework 2**due Tuesday 10/5.

**Thu 8/26:**overview; symplectic vector spaces, standard basis, subspaces; symplectic manifolds.**Tue 8/31:**de Rham cohomology, Lie derivative; symplectic form on the cotangent bundle, Lagrangian submanifolds**Thu 9/2:**Hamiltonian vector fields, Hamiltonian diffeomorphisms vs. symplectomorphisms, flux; symplectic isotopy, Moser's theorem.**Tue 9/7:**Moser and Darboux theorems, local Moser theorem.**Thu 9/9:**Weinstein's Lagrangian neighborhood theorem, and consequences.**Tue 9/14:**Hamiltonian group actions, moment maps; symplectic toric manifolds.**Thu 9/16:**symplectic reduction; contact manifolds.**Tue 9/21:**contact manifolds continued; complex structures and compatibility.**Thu 9/23:**almost-complex structures, compatible triples, contractibility; vector bundles and connections**Tue 9/28:**curvature, Chern classes**Thu 9/30:**constraints on almost-complex 4-manifolds; types of vectors and differential forms, splittings; integrability.**Tue 10/5:**Nijenhuis tensor and integrability; Kähler manifolds; CP^n.**Thu 10/7:**pseudoholomorphic curves; Gromov's non-squeezing theorem, sketch of proof, monotonicity formula.**Tue 10/12:**local behavior of pseudoholomorphic curves; linearized dbar operator.**Tue 10/19:**moduli space of J-holomorphic curves; ellipticity; transversality.**Thu 10/21:**bubbling, stable maps, and Gromov compactness; Gromov-Witten invariants.**Tue 11/2:**multiple covers and transversality; the algebraic approach to GW theory; existence of J-spheres in S^2xS^2.**Thu 11/4:**Morse theory: Morse complex, Morse-Smale condition, d^2=0, continuation maps.**Tue 11/9:**Isomorphism between Morse homology and cellular homology; Arnold conjecture, action functional, Floer's equation.**Tue 11/16:**Hamiltonian Floer homology continued: differential, d^2=0, bubbling, independence of H, isomorphism with Morse homology in the monotone case.**Thu 11/18:**Lagrangian Floer homology: motivation, differential, Maslov index.**Tue 11/23:**Lagrangian Floer homology: compactness, bubbling, d^2=0 in the absence of bubbling.**Tue 11/30:**Lagrangian Floer homology: bubbling of discs and obstruction; Hamiltonian isotopy invariance.**Thu 12/2:**Lagrangian Floer homology: graded lifts; Floer homology in the cotangent bundle, relation to Morse theory; the monotone case, Oh spectral sequence.

The course will provide an overview of symplectic topology. It will start with fairly standard material, to be followed by a brief introduction to some more advanced topics (Floer homology, and constructions of symplectic manifolds). The main topics to be covered include:

- Linear symplectic geometry.
- Symplectic manifolds; symplectomorphisms; Lagrangian submanifolds.
- Darboux and Moser theorems, Lagrangian neighborhood theorem.
- Contact manifolds.
- Complex vector bundles, Chern class.
- Almost-complex structures, compatibility, integrability.
- Kähler manifolds.
- Hamiltonian group actions, moment maps and symplectic quotients.
- Pseudoholomorphic curves.
- Floer homology (a brief introduction).
- Constructions of symplectic manifolds (blowups, connected sums, fibrations, surgeries).

**Prerequisites:** Math 214 (Differentiable Manifolds) and 215A (Algebraic
Topology)

**Grading:** based on homework (every two weeks)

- Most of material we'll cover can be found in:
**D. McDuff and D. Salamon**,*Introduction to Symplectic Topology*, Oxford Mathematical Monographs, 2nd edition. - Another reference for the first part of the course is:
**A. Cannas da Silva**,*Lectures on Symplectic Geometry*, Lecture Notes in Mathematics 1764, Springer-Verlag.

*Note: the university library has this text available as an e-book here (from campus only); you can order a deeply discounted soft-bound printed copy from Springer from the e-book download page.*