Math 253y - Symplectic Manifolds and Lagrangian Submanifolds - Fall 2018

D. Auroux - Tue. & Thu., 10:30-11:45am, Science Center 222

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

Office: Science Center 539.
Office hours: Tuesdays 12-1 and Thursdays 9-10 (subject to change).
Course assistant: Yu-Wei Fan (ywfan@math), office hours Wednesdays 1:30-3 in SC 310.

Announcements

• Nov 3: There will be no class on Thursday Nov 15. We will make up by having an extra lecture during reading period.
• Nov 3: Homework 3 is here. It'll be the last assignment.
• Oct 15: please take a moment to fill our midterm course questionnaire

Homework

• Homework 1 due Thu 9/27.
• Homework 2 due Thu 10/18.
• Homework 3 due Tue 12/4.

Material covered

• Tue 9/4: overview of the course: symplectic manifolds, Lagrangian submanifolds, examples, results and open problems.
• Thu 9/6: (a) overview continued: J-holomorphic curves, Lagrangian Floer homology, Fukaya category. (b) basic symplectic geometry: symplectic manifolds, Lagrangian submanifolds.
• Tue 9/11: Hamiltonian vector fields; Moser and Darboux theorems.
• Thu 9/13: Lagrangian neighborhood theorem; Hamiltonian actions, moment maps.
• Tue 9/18: Atiyah-Guillemin-Sternberg convexity theorem; Delzant classification of toric symplectic manifolds.
• Thu 9/20: Symplectic reduction, examples.
• Tue 9/25: Constructing Lagrangians: reduction and fibrations.
• Thu 9/27: Constructing Lagrangians: Lefschetz fibrations; generating functions.
• Tue 10/2: Constructing Lagrangians: h-principle, spinning, and surgery.
• Thu 10/4: Lagrangian fibrations and locally integrable systems, action-angle coordinates.
• Tue 10/9: Lagrangian fibrations: examples.
• Thu 10/11: Almost-complex structures, integrability.
• Tue 10/16: Kähler potentials; J-holomorphic curves.
• Thu 10/18: The linearized Cauchy-Riemann operator and its index.
• Tue 10/23: Fredholm index and Maslov index; moduli spaces.
• Thu 10/25: Regularity criteria.
• Tue 10/30: Non-regularity and perturbations; introduction to compactness.
• Thu 11/1: Gromov compactness and stable curves.
• Tue 11/6: Gromov-Witten invariants, quantum cohomology.
• Thu 11/8: Disc-counting invariants; superpotential.
• Tue 11/13: Superpotential: examples; wall-crossing.
• Thu 11/15 no class
• Tue 11/20: Lagrangian Floer cohomology: definition, index and grading.
• Tue 11/27: Lagrangian Floer cohomology: d^2=0, invariance, HF(L,L).
• Thu 11/29: Lagrangian Floer theory: product, A_∞ structure.
• Tue 12/4: Fukaya category; mapping cones and generators; example: T^2.
• Thu 12/6: Homological mirror symmetry for T^2.

Lecture notes

• Course overview
• 1. Symplectic manifolds and neighborhood theorems
• 2. Moment maps and symplectic reduction
• 3. Constructing Lagrangians
• 4. The geometry of Lagrangian fibrations
• 5. Almost-complex structures
• 6. J-holomorphic curves
• 7. Gromov compactness and stable maps
• 8. Disc enumerative invariants
• 9. Lagrangian Floer cohomology and Fukaya categories: §1-3 of A beginner's introduction to Fukaya categories

Course outline

The course will start with a review of standard symplectic topology: symplectic manifolds, Lagrangian submanifolds, neighborhood theorems, almost-complex structures and compatibility, Hamiltonian group actions. The focus will then shift towards J-holomorphic curves: moduli spaces, Gromov compactness, etc., with a view towards Lagrangian Floer theory. The final part of the course will give a taste of more advanced topics: invariants of monotone Lagrangians; Fukaya categories; mirror symmetry.

Provisional list of topics (to be adjusted):

• Symplectic manifolds; symplectomorphisms; Lagrangian submanifolds. Darboux and Moser theorems, Lagrangian neighborhood theorem.
• Hamiltonian group actions, moment maps and symplectic quotients. Toric manifolds.
• Constructions of Lagrangian submanifolds.
• Almost-complex structures, compatibility, integrability.
• Pseudoholomorphic curves, transversality, Gromov compactness.
• Lagrangian submanifolds and their enumerative geometry.
• Lagrangian Floer homology and Fukaya categories.
• Introduction to homological mirror symmetry.

Prerequisites: a solid knowledge of differential geometry, and basic algebraic topology (Math 230a and Math 231a).

Reference books

Basic symplectic geometry:

• D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs.
• 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.

J-holomorphic curves:

• D. McDuff, D. Salamon, J-holomorphic curves and symplectic topology, AMS Colloquium Publ. 52, AMS, 2004.

Floer homology and Fukaya categories:

• P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Math. Soc., Zürich, 2008.
• D. Auroux, A beginner's introduction to Fukaya categories, arXiv:1301.7056.