## Mathematics 790, Spring 2017

### Minicourse: A beginner's guide to A-infinity categories and Fukaya categories

**
Time: Tuesdays and Thursdays, 10:05-11:20am**

Dates: February 14 to March 21

Location: Physics 205

Instructor: Lenny Ng

Fukaya categories play a crucial role in modern symplectic topology
and provide one side of Kontsevich's famous "Homological Mirror
Symmetry" conjecture (the other side is given by coherent sheaves). In
symplectic topology, Fukaya categories arose from the pioneering work
of Floer on Lagrangian intersection Floer theory, which also underlies
many recent developments in low-dimensional topology (e.g., Heegaard
Floer theory).

In this minicourse, I will give a (rather nontechnical) introduction
to Fukaya categories. This includes a geometry-flavored overview of
Lagrangian intersection Floer theory as well as an algebra-flavored
treatment of the algebraic structures necessary to set up Fukaya
categories, including A-infinity algebras and A-infinity categories. My
intention is to make this minicourse as accessible as possible.
Familiarity with algebraic topology and differential geometry at the
beginning graduate level (Math 611 and 621) will be helpful;
familiarity with symplectic geometry will not be assumed.

Here is an outline of topics I plan to address, to varying extents, in the minicourse:

- 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.

I will loosely follow Denis Auroux's "A beginner's introduction to Fukaya categories" but will supplement this with other sources.

Course notes (will update as we go); for the early material on Morse homology, there are also notes from my previous minicourse in spring 2015.