Math 257B: Topics in Symplectic Geometry

Aspects of Fukaya categories

Monday, Wednesday (and occasionally Friday) 1:30-2:50pm in 240-110

Teaching Staff

Instructor Sheel Ganatra
Office 380-382E
e-mail ganatra (at) math (dot) stanford (dot) edu

Course Description and Prerequisites

This course aims to introduce the Fukaya category of a symplectic manifold, an intricate categorical invariant built out of Lagrangian submanifolds and their Floer homology groups (built out of moduli spaces of J-holomorphic discs with Lagrangian boundary conditions), and the symplectic half of Kontsevich's homological mirror symmetry conjecture.

In fact, the course will describe several recent flavors of Fukaya categories (such as compact, wrapped, and infinitesimal Fukaya categories) associated to different geometric settings. A substantial part of the course will focus on understanding Fukaya categories associated to singular symplectic fibrations (such as Lefschetz fibrations), which give both non-trivial symplectic invariants of singularities and a formalism for interpolating between Fukaya categories of the general fiber and the total space. This portion of the course will draw upon new structures developed in joint work with Abouzaid, as well as recent work of Seidel and Abouzaid-Seidel.

We will assume some familiarity with basic results in symplectic geometry. However, no experience with the Fukaya category, J-holmorphic discs, or Floer-theoretic methods will be assumed.

Announcements

References and resources

We will not use a specific text for this course. Below are some general references on Fukaya categories, Lagrangian (and Hamiltonian) Floer homology, and J-holomorphic curves.

Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Day Lecture topics Lecture notes Additional References
3/28 Course Overview: Lagrangian Floer homology and Fukaya categories at a glance. Flavors of the Fukaya category and their relations. Notes by Cedric De Groote.
3/30 J-holomorphic curves and the energy identity. Lagrangian Floer Homology: The action functional and its critical points and flowlines. Spaces of J-holomorphic strips/discs and the Floer differential. Notes by Cedric De Groote.
4/1 Lagrangian Floer homology II: Fredholm theory and transversality. The Maslov index. Gromov's compactness theorem. Notes by Cedric De Groote.
4/6 Lagrangian Floer homology III: More Gromov compactness, d^2 = 0 (and when this fails). Hamiltonian isotopy invariance. Notes by Cedric De Groote.
4/8 Lagrangian Floer homology IV: Floer's equation versus pseudoholomorphic curves. Hamiltonian isotopy invariance via continuation maps. Relation to Morse homology, Oh and Pozniak spectral sequences. Notes by Cedric De Groote.
4/11 Absolute and relative gradings on Lagrangian Floer homology. Product structures. Notes by Cedric De Groote.
4/13 Orientations of moduli spaces. Determinant lines and the appearance of Spin (or relatively Pin) structures in Lagrangian Floer homology. Floer homology with signs. Notes by Cedric De Groote.
4/18 More orientations. The product in Lagrangian Floer homology, which is composition (cohomologically) in the Donaldson Fukaya category. Identity morphisms. Notes by Cedric De Groote.
4/25 A-infinity structures and the full Fukaya category. Notes by Cedric De Groote.
4/27 A-infinity functors and quasi-isomorphisms. The statement of invariance for the Fukaya category. Properties of A-infinity structures and their presence in topology: homological perturbation/transfer, quasi-isomorphisms are invertible, etc. Massey products as an example of invariants not arising on the cohomology. Formality and non-formality. Notes by Cedric De Groote.
5/2 (Shorter class): Miscellaneous topics: cohomologically unital versus strictly unital versus homotopy unital A-infinity categories, Floer cohomology and Fukaya category for objects with local systems, and curved A-infinity categories. A case in which spheres and discs exist but there are still classical methods available: monotone symplectic manifolds and their monotone Lagrangians. Notes by Cedric De Groote.
5/4 More monotone Floer homology: various holomorphic curve theory miracles when (X,L) is monotone. The "charge" (or curvature) of a monotone Lagrangian, the decomposition of the monotone Fukaya category into summands indexed by charge. More general frameworks for discussing curvature: FOOO's Morse-Bott framework for Floer cohomology, and Cornea-Lalonde's cluster homology. Notes by Cedric De Groote.
5/9 (Shorter class) An example: The clifford torus in CP^n. Lagrangian connect sums and long-exact sequences of Floer cohomology groups. The more general notion of exact triangles in A-infinity categories. Pre-triangulated A-infinity categories. Notes by Cedric De Groote.
5/11 The pre-triangulated (and split-closed pre-triangulated) hull of an A-infinity category via A-infinity modules and the Yoneda embedding. Generation and split-generation in the Fukaya category. Notes by Cedric De Groote.
5/16 Categorical localizations and quotients. Symplectic Landau-Ginzburg (LG) models and Floer cohomology of admissible Lagrangians in a symplectic LG model. Examples: Lefschetz (and Lefschetz-Bott) fibrations, and their thimbles (and generalized thimbles). Notes by Cedric De Groote.
5/18 Abouzaid-Seidel's construction of the Fukaya category of a Landau-Ginzburg model as a categorical localization (generalizing Seidel's construction of Fukaya categories of Lefschetz fibrations). Properties and examples. Notes by Cedric De Groote.
5/23 Canonical functors on the Fukaya category of a Landau-Ginzburg model, the Fukaya category of a fibre, and between the two categories: once wrapping, monodromy, intersection with the fiber, and the Orlov functor. Exact triangles relating these functors (a statment) and applications: another proof of Seidel's LES of a Dehn twist. Towards explaining some of these functors: a localization model of the Fukaya category of the general fibre. Notes by Cedric De Groote.
5/25 Miscellanous remarks about split-generation and projection onto a subcategory. Serre functors on Fukaya categories of Landau-Ginzburg models and the wrapped Fukaya category. The exact triangle for once wrapping and generation of Landau-Ginzburg categories by Orlov Lagrangians (when wrapped Floer homology vanishes). Notes by Cedric De Groote.
6/1 Generation criteria for Fukaya categories. Notes by Cedric De Groote.
6/3 Generating Fukaya categories of Landau-Ginzburg models. Notes by Cedric De Groote.