Mathematics 790 (790-90-03), Fall 2023

Minicourse: Introduction to Lagrangian Floer theory

Time: TuTh 1:25 - 2:40 pm
Dates: September 21* - October 31
Location: Physics 227
Instructor: Lenny Ng

This minicourse will be 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. I will not assume familiarity with symplectic geometry. It will be quite helpful to know something about Morse homology; I'll do a lightning review in the minicourse, but you will be much better served if you do some reading about Morse homology in advance of the minicourse, along the lines of the notes from another of my past minicourses:

Lecture notes from my fall 2018 Morse homology minicourse (ignore the 2015 date on the first page of the notes; these are from 2018).

I've taught variants on this Floer minicourse previously, in fall 2020 and spring 2017. Here's a more detailed outline of topics I plan to cover:

• 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.

For your convenience, here are my lecture notes from the spring 2017 minicourse, which I'll also loosely be following.