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MAT 682. Advanced Topics in Differential Geometry: The Fukaya Category

Course Instructor: Mark McLean (markmclean AT math.stonybrook.edu)

Monday, Wednesday 2:30pm-3:50pm, Physics P125

Introduction to the Course

The aim of this course is to introduce an invariant of a symplectic manifold called the Fukaya category and to describe a few applications of this invariant. This is a `category' whose objects are built from Lagrangian submanifolds and whose morphisms come from intersection points of these Lagrangians along with additional data. In the first part of the course we will define Lagrangian Floer cohomology and give some dynamical applications of this invariant. After that we will define the Fukaya category in the simplest setting and state some additional properties. At the end of the course we will explain a couple of applications and also sketch some of the basic ideas behind homological mirror symmetry. This is an advanced course.

Prerequisites

You need to know some symplectic geometry and some complex geometry.

Office Hours:

Suggested Reading

Symplectic Geometry
McDuff-Salamon: Chapters 1,3,4 (use Chapter 2 as a reference).
Morse Homology
Hutchings: Lecture notes on Morse homology (with an eye towards Floer theory (online).

Audin Damian: Morse Theory and Floer Homology (this has more detail than Hutchings notes).

Lagrangian Floer Cohomology
Floer: Morse theory for Lagrangian intersections
Fukaya Categories
Auroux: A beginner's introduction to Fukaya categories

Smith: A symplectic prolegomenon

Seidel: Fukaya Categories and Picard-Lefschetz Theory. (Skip chapter I part 6 at least at the beginning. Also it might be good to read chapter 1 part 1 and then go to part II and use part I as a reference.)

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