Mathematics 790, Fall 2020

Minicourse: A gentle introduction to Floer theory in symplectic geometry

Time: MW 10:15-11:30 am
Dates: Wednesday September 16 to Wednesday October 14
Location: online on Zoom; enter meeting ID 920 4953 2379
Instructor: Lenny Ng

The minicourse is now finished. Thanks everyone who attended! Lecture notes and password-protected lecture recordings are available below.

This minicourse will be taught entirely online, on Mondays and Wednesdays during the second third of the fall 2020 semester.

The Zoom meeting ID above will work for each meeting of the minicourse. You will need a password to access the Zoom meetings; please contact me for the password if you'd like to request access. The minicourse is mainly intended for members of the Duke community and nearby areas, and you may have already received an email from me with the password.

Here are the lecture notes (in the form of a PDF of the screens that I share during the class) and links to the Zoom recordings of the classes. Please note that the password for the Zoom recordings is the password for the minicourse, but with the five characters 2020! added to the end.

Lecture notes part 1 (9/16 through 9/28)
Lecture notes part 2 (9/30 through 10/5)
Lecture notes part 3 (10/7 through 10/14)
9/16 lecture
9/21 lecture
9/23 lecture
9/28 lecture
9/30 lecture
10/5 lecture
10/7 lecture
10/12 lecture
10/14 lecture

Here are links to notes from two of my previous minicourses. I will be following the notes from the Fukaya category minicourse fairly carefully in the present minicourse, so it may be helpful to have that handy to follow along during lectures.

A-infinity categories and Fukaya categories (spring 2017)
Morse theory and Floer homology (spring 2015): see this for background on Morse homology.

My plan for this minicourse is to present a rather nontechnical introduction to Lagrangian intersection Floer theory, which underlies many recent developments in symplectic topology as well as low-dimensional topology (e.g. through Heegaard Floer theory). This includes a very quick overview/review of Morse homology building to a geometry-flavored overview of Lagrangian Floer theory; an introduction to associated algebraic structures such as A-infinity algebras and Fukaya categories; and probably some cursory discussion of analytical issues involved in all of these constructions.

My intention is to make this minicourse as accessible as possible. I will assume familiarity with smooth manifolds at the level of Math 620, and algebraic topology at the level of Math 611. It will also be helpful (but not absolutely necessary) to know something about Morse homology. I will not assume familiarity with symplectic geometry.

I taught a variant of this minicourse in Spring 2017. The outline that I posted for that minicourse is also a good outline for the present one:

Morse homology (the basic setup)
background in symplectic geometry: symplectic forms, Lagrangian submanifolds, Hamiltonian diffeomorphisms
Lagrangian intersection Floer homology
A-infinity algebras and A-infinity categories
Fukaya categories.