Symplectic methods in low-dimensional topology: Math 566


Meeting time: T Th 11-12:20
Classroom: Fine 1201
Instructor: Peter Ozsváth
phone: 609-258-4222
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):

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 three-manifolds.
    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 1-39.)
  • Lectures on Symplectic Geometry A. Cannas da Silva.
  • Handout.
    Lecture 8 (Sept 28): The Morse-Smale 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 three-manifolds 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 three-manifold 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 four-manifolds 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.