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:

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.