18.979 Graduate Geometry Seminar Spring 2008:
The symplectic category, holomorphic curves, quilts, and topological applications
Lecture Notes:
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
Literature:
Dusa McDuff and Dietmar Salamon,
Introduction to Symplectic Topology ,
Oxford University Press, April 1995.
Dusa McDuff and Dietmar Salamon,
J-Holomorphic Curves and Symplectic Topology ,
AMS Colloquium Publications, Vol. 52, 2004.
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, Morse Homology, Birkhaeuser 1993.
Victor Guillemin and Shlomo Sternberg, The moment map revisited ,
J.Diff.Geom. 69(1), 137-162, 2005.
Katrin Wehrheim and Chris Woodward, Functoriality for Lagrangian correspondences in Floer theory, preprint.
Schedule: Tue/Thu 3-4:30 in 2-131 ; office hours Tue/Thu 2-3 in 2-277
Outline:
This lecture course is aimed at beginning graduate students
in geometry as well as more advanced researchers.
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--x N in
a product of symplectic manifolds M and N.
These are viewed as 'correspondence' from M to N; generalizing the
notion of a symplectomorphism (where M and N are the same).
I plan to cover the following topics:
- 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
Prerequisites:
geometry of manifolds (18.965), some differential analysis (18.155), some symplectic topology (18.966).
Taking 18.966 parallel should be possible, but please contact me in advance.
Previous exposure to holomorphic curves is useful but not necessary.
Contact: Katrin Wehrheim
( katrin (guess what) math.berkeley.edu )