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
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.