Symplectic methods in lowdimensional topology: Math 566
Meeting time: T Th 1112:20
Classroom: Fine 1201
Instructor: Peter Ozsváth
phone: 6092584222
email:
petero@math.princeton.edu
office: 1106
office hrs: TBA.
The course:
This course will begin as an introduction to
Heegaard Floer homology.
Recommended reading (more is included in the Course plan below):

J. Milnor Morse Theory. Everyone should know this.

Ana Canas da Silva Lectures on Symplectic Geometry. Basic background on symplectic geometry.

R. Gompf and A. Stipsicz
4Manifolds and Kirby Calculus, especially Chapter 2. Background on (4dimensional) differential topology.

M. Audin and M. Damian Morse theory and Floer homology. This is useful background for Floer homology.

P. S. Ozsváth, A. Stipsicz Z. Szabó Grid homology for knots and links.
A very gently introduction to a combinatorial model for knot Floer homology.
I will also hand out some supplemental material on Heegaard Floer homology.
Grading:
The course grade is calculated as follows:
Homework assignments:
There will be 5 homework assignments (with equal weights) throughout the semester.
Homework 1, due Tues, Sept 19th.
Exercise 4.3.8 from Grid homology for knots and links.
Exercise 4.4.3.
Exercise 4.6.6
Exercise 4.8.2
Homework 2, due on Oct 5th.
Exercise 1.2.4 from handout 1.
Exercise 2.3.4
Exercise 2.4.3
Exercise 3.1.4
Homework 3 is on Heegaard Floer homology. It is due on Oct 31.
Homework 4 is on further properties of Heegaard Floer homology. It is due Nov 13th
Homework 5 is on bordered Heegaard Floer homology. It is due Dec 5.
Final Exam:
There will be an oral final exam (exact time
should be set with me in advance) about some topic covered in lectures.
Course plan
Here is a rough plan for the course. This plan will adapt to the interests and needs of the audience.
For example, we could take a more rapid course towards knot invariants; or we could spend a little more time on the algebraic underpinnings of bordered Floer homology.
Lecture 1 (Sept 5): Introduction to the subject.
Recommended reading:
An introduction to Heegaard Floer homology
by Szabó and me.
Lectures on Heegaard Floer homology
by Szabó and me.
Handout. This is Handout #1 in ``Modules'' for the Canvas page for our course.
Part 1: Grid diagrams and knot Floer homology
Lecture 2 (Sept 7): Grid diagrams representing knots, and their chain complexes.
Recommended reading:
A combinatorial description of knot Floer homology
by Ciprian Manolescu, Sucharit Sarkar, and me.
On combinatorial link Floer homology
by Ciprian Manolescu, Z. Szabó, D. Thurston, and me.
Grid homology for knots and links by A. Stipsicz, Z. Szabó and me.
(See especially Chapters 1,3,4.)
Lecture 3 (Sept 12): Grid homology is a knot invariant.
Recommended reading:
Grid homology for knots and links,
Chapters 5 and 7.
Lecture 4 (Sept 14): Some applications of grid homology.
Recommended reading:
Grid homology for knots and links by A. Stipsicz, Z. Szabó and me.
Chapters 6 and 8.
Part 2: Topological background
Lecture 5 (Sept 19): Heegaard diagrams for threemanifolds.
Recommended reading:
Handout (Chapters 1 and 2).
Lecture 6 (Sept 21): Symmetric products.
Recommended reading:
Symmetric products of an algebraic curve
I. G. MacDonald, Topology 1962.
Handout.
Part 3: An outline of Lagrangian Floer homology
Lecture 7 (Sept 26): Fundamental notions: fundamentals of Morse theory and symplectic geometry.
Recommended reading:
Morse Theory John W. Milnor. (See esp Part 1, pp 139.)
Lectures on Symplectic Geometry
A. Cannas da Silva.
Handout.
Lecture 8 (Sept 28): The MorseSmale complex.
Recommended reading:
Lecture notes on Morse homology Michael Hutchings
Handout.
Lecture 9 (Oct 3): Floer homology.
Recommended reading:
Morse Theory and Floer homology M. Audin and M. Damian
Handout.
Lecture 10 (Oct 5): Floer homology, continued.
Part 4: The construction of Heegaard Floer homology
Lecture 11 (Oct 10): The Maslov index formula.
Recommended reading:
Handout.
Lecture 12 (Oct 12): The definition of Heegaard Floer homology.
Recommended reading:
Holomorphic disks and topological invariants for closed threemanifolds
by Szabó and me.
Handout.
Lecture 13 (Oct 17): Holomorphic triangles.
Lecture 14 (Oct 19): Invariance of Heegaard Floer homology.
Lecture 15 (Oct 31): Example computations.
Part 5: Properties of Heegaard Floer homology
Lecture 16 (Nov 2): The exact triangle.
Recommended reading:
Holomorphic disks and threemanifold invariants: properties and applications
by Szabó and me.
Handout.
Lecture 17 (Nov 7): Applications of the exact triangle.
topological invariants for closed
Absolutely graded Floer homologies and intersection forms for fourmanifolds with boundary
by Szabó and me.
Lecture 18 (Nov 9): Knot Floer homology revisited.
Recommended reading:
An overview of knot Floer homology
Holomorphic disks and knot invariants
Lecture 19 (Nov 14): Surgery formulas.
Part 6: Bordered Floer homology
Lecture 20 (Nov 16): Bordered Floer homology: algebraic background.
Recommended reading:
Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology by R. Lipshitz, D. P. Thurston, and me.
Bordered Heegaard Floer homology by R. Lipshitz, D. P. Thurston, and me.
Lecture 21 (Nov 28): Bordered Floer homology.
Lecture 22 (Nov 30): The pairing theorem.
Lecture 23 (Dec 5): Bordered knot Floer homology
Recommended reading:
Algebras with matchings and knot Floer homology
by Z. Szabó and me.
Lecture 24 (Dec 7): Recent developments.