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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.
You need to know some symplectic geometry and some complex geometry.
- Monday 1pm-2pm (Math 4-114)
- Tuesday 11:30am-12:30pm (Math 4-114).
- Wednesday 1pm-2pm (Math 4-114)
McDuff-Salamon: Chapters 1,3,4 (use Chapter 2 as a reference).
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
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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