Lecture 1 - linear symplectic category, composition of Lagrangian correspondences

Lecture 2 - general composition and examples of Lagrangian correspondences

Lecture 3 - Lagrangian correspondences arising from moment maps

Lecture 4 - generalized Lagrangian correspondences

Lecture 5 - generalized Floer homology

Lecture 6 - generalized Floer homology - trajectories

Lecture 7 - generalized Floer homology - quilts and how to fold them

Lecture 8 - quilted Floer homology - perturbations and monotonicity

Lecture 9 - quilted Floer homology - compactness

Lecture 10 - quilted Floer homology - invariance

Lecture 11 - quilted Floer homology and geometric composition

Lecture 12 - pseudoholomorphic quilts

Lecture 13 - quilt invariants

Lecture 14 - the symplectic 2-category

Lecture 15 - Floer field theory

Lecture 16 - compactness - local estimates

Lecture 17 - compactness - mean value inequalities

and a general pamphlet on this trick: Energy quantization and mean value inequalities ...

Lecture 18 - compactness - up to energy concentration

Lecture 19 - compactness - bubbling

Lecture 20 - bubbling at seams, and the figure eight bubble

Dusa McDuff and Dietmar Salamon,

Dusa McDuff and Dietmar Salamon,

Dietmar Salamon, Lectures on Floer homology, Lecture Notes for the IAS/PCMI Graduate Summer School on Symplectic Geometry and Topology, Preprint, December 1997.

Matthias Schwarz,

Victor Guillemin and Shlomo Sternberg,

Katrin Wehrheim and Chris Woodward, Functoriality for Lagrangian correspondences in Floer theory, preprint.

A basic introduction to some aspects of symplectic topology and holomorphic curves will lead into recent research of Chris Woodward and myself. The main object of the course will be Lagrangian submanifolds L ⊂ M

- Lagrangian correspondences and their geometric composition

- sequences of Lagrangian correspondences, building a symplectic category

- Hamiltonian group actions with moment maps, and Lagrangian correspondences arising from these

- holomorphic curves with Lagrangian boundary conditions

- Floer theory and morphisms on Floer homology arising from holomorphic curves with cylindrical ends

- the Donaldson-Fukaya category of a symplectic manifold

- holomorphic quilts, using Lagrangian correspondences as boundary conditions

- generalized Floer homology and morphisms arising from quilts

- isomorphism of Floer homologies under composition of Lagrangian correspondences

- adiabatic analysis for shrinking strips in quilts, and the mysterious figure eight bubble

- the extended Donaldson category and functors arising from Lagrangian correspondences

- a symplectic 2-category, using holomorphic quilts

- a general framework for constructing topological invariants via a decomposition into simple pieces and a representation in the symplectic category

- examples of invariants for knots and 3-manifolds arising from moduli spaces of bundles

- construction of a 2+1+1 category-valued topological quantum field theory, factoring through the symplectic 2-category