Math 242 - Symplectic Geometry - Fall 2010

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

Instructor: Denis Auroux (auroux@math.berkeley.edu)

Office: 817 Evans.
Office hours: Mondays 4-6, Tuesdays 5-6 (to be confirmed).

Homework

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

Material covered

• 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.

Course outline

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)

Textbooks

• 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.