Fukaya Categories of Surfaces, Math 595 FSC, Fall 2022

• Lectures: MWF 11:00–11:50am in 441 Altgeld Hall.
• Instructor: James Pascaleff
  • Email: jpascale@illinois.edu; Office: 357 Altgeld Hall; Phone: (217) 244-7277.
  • Office hours: Wednesdays 12:00–2:00pm.
• Course website: https://pascaleff.github.io/595fa22/

The Fukaya category is a sophisticated invariant of symplectic manifolds whose definition in full generality is intricate. In the original formulation, objects of this category are certain “branes” supported on Lagrangian submanifolds, and the composition laws in the category involve pseudo-holomorphic maps with Lagrangian boundary conditions (Floer theory).

At the same time, a category, like any other algebraic structure, can admit many presentations. For instance, we could try to find a presentation of the Fukaya category by generators and relations, or we could try to build the Fukaya category of a symplectic manifold up from categories associated to smaller pieces of the manifold.

Let \(S\) be a oriented surface (2-dimensional real manifold). The Fukaya category of \(S\) is now well-understood enough that we know multiple ways of presenting it. The goal of this course is to study the Fukaya category of \(S\) via multiple presentations, the original Floer theoretic definition being only one of them. Others include the theory of “gentle algebras” (by results of Haiden-Katzarkov-Kontsevich), and a perspective that views the Fukaya category of \(S\) as an object in homological algebra (Dyckerhoff-Kapranov).

Thus, although this course will include some Floer theory, it will not be strictly speaking necessary in order to understand all of the different approaches to the Fukaya category of a surface. My hope is that students with different backgrounds in geometry and algebra will be able to find at least one way of thinking about the Fukaya category that makes sense to them. Applications of these ideas to Homological Mirror Symmetry will also be discussed.

• Prerequisites: Some familiarity with the tools of algebraic topology is strongly recommended.

• DG categories
  • A. I. Bondal and M. M. Kapranov, Enhanced Triangulated Categories.
  • B. Keller, On differential graded categories.
• \(A_\infty\) categories
  • P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Chapter I.
  • B. Keller, Introduction to A-infinity algebras and modules.
  • G. Faonte, A-infinity functors and homotopy theory of dg-categories.
• Associahedra and moduli spaces of disks
  • P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Chapter II, Section 9.
  • Related: T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mappings.
• Gradings and signs on surfaces
  • P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Chapter II, Section 13.
• Formal enlargements
  • P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Chapter I.

1. Motivation for the Fukaya categories of surfaces.
2. DG categories.
3. \(A_\infty\) categories.
4. The Associahedron.
5. Polygons and moduli spaces of disks.
6. Operads.
7. Towards the Fukaya category of a surface.
   • Background on the conformal automorphisms of the disk.
8. First examples of computations in the Fukaya category.
   • For the example of triangles on the two-torus and the relation to theta functions, see these notes from a previous 595 course.
9. Gradings and signs on surfaces.
10. Verifying the \(A_\infty\) associativity equations on surfaces. (Notes under construction)
11. Formal enlargements of \(A_\infty\) categories.