Math 253y  Symplectic Manifolds and Lagrangian Submanifolds  Fall 2018
D. Auroux 
Tue. & Thu., 10:3011:45am, Science Center 222
Instructor:
Denis Auroux (auroux@math.harvard.edu)
Office: Science Center 539.
Office hours: Tuesdays 121 and Thursdays 910 (subject to
change).
Course assistant: YuWei Fan (ywfan@math), office hours Wednesdays 1:303 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
There will be 34 homework assignments during the semester, including
a more substantial assignment at the end of the semester serving as
takehome final assessment.
Material covered
 Tue 9/4: overview of the course: symplectic manifolds,
Lagrangian submanifolds, examples, results and open problems.
 Thu 9/6: (a) overview continued: Jholomorphic 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: AtiyahGuilleminSternberg 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: hprinciple, spinning, and surgery.
 Thu 10/4: Lagrangian fibrations and locally integrable systems, actionangle coordinates.
 Tue 10/9: Lagrangian fibrations: examples.
 Thu 10/11: Almostcomplex structures, integrability.
 Tue 10/16: Kähler potentials; Jholomorphic curves.
 Thu 10/18: The linearized CauchyRiemann operator and its index.
 Tue 10/23: Fredholm index and Maslov index; moduli spaces.
 Thu 10/25: Regularity criteria.
 Tue 10/30: Nonregularity and perturbations; introduction to compactness.
 Thu 11/1: Gromov compactness and stable curves.
 Tue 11/6: GromovWitten invariants, quantum cohomology.
 Thu 11/8: Disccounting invariants; superpotential.
 Tue 11/13: Superpotential: examples; wallcrossing.
 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
These handwritten notes may be incomplete or incorrect  use at your
own risk.
Course outline
The course will start with a review of standard symplectic topology:
symplectic manifolds, Lagrangian submanifolds, neighborhood theorems,
almostcomplex structures and compatibility, Hamiltonian group actions.
The focus will then shift towards Jholomorphic 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.
 Almostcomplex 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, SpringerVerlag.
Note: the university library has this text available as an ebook
here.
Jholomorphic curves:

D. McDuff, D. Salamon, Jholomorphic curves and symplectic
topology, AMS Colloquium Publ. 52, AMS, 2004.
Floer homology and Fukaya categories:
 P. Seidel, Fukaya categories and PicardLefschetz
theory, European Math. Soc., Zürich, 2008.

D. Auroux, A beginner's introduction to Fukaya categories,
arXiv:1301.7056.