18.979 Graduate Geometry Seminar Spring 2007:
Polyfold-Fredholm theory and generalized Lagrangian Floer theory
Hofer-Wysocki-Zehnder have recently introduced new "scale" smooth
structures on Banach spaces and "polyfolds", a generalization of
Banach manifolds, allowing for locally varying dimensions. In this
language, a compactified moduli space can be described as the zero set
of a "Fredholm section" in a "polyfold bundle". Examples of such
moduli spaces are
the set of Morse flow lines, including broken trajectories; or the set
of holomorphic curves, including nodal curves. More generally, the
formation of singularities can be described in the same category that
nonsingular objects lie in. Two main features of this
Polyfold-Fredholm theory are an abstract transversality theorem and an
implicit function theorem.
I will develop the Polyfold-Fredholm theory from scratch and
illustrate its application at a setup for generalized Lagrangian Floer
theory (overcoming obstructions from disk bubbles).
The latter is a finitely generated version of the Fukaya-Oh-Ono-Ohta
$A_\infty$ algebra, proposed by Cornea-Lalonde. It requires the study
of moduli spaces of trees of holomorphic disks, connected by Morse
flow lines.
Prerequisites:
basic topology, differential geometry, and functional analysis
(Previous exposure to moduli spaces of PDE's is useful but not necessary.)
Literature:
Overview (CDM):
H.Hofer, A General Fredholm Theory and Applications
(download).
Script part 1: H.Hofer,K.Wysocki,E.Zehnder,
A General Fredholm Theory I: A Splicing-Based Differential Geometry
A preliminary version of
The Book
Preprint on generalized Lagrangian Floer theory via cluster moduli spaces:
O.Cornea,F.Lalonde,
Cluster Homology
Lecture notes:
I am planning to produce lecture notes on the application to
Lagrangian Floer theory.
If you are interested in helping with this project let me know -
funding might be available.
Schedule: MWF 11-12 (but will poll on rescheduling after first week)
office hours: MWF 12-1 at lunch
Contact: Katrin Wehrheim
( katrin (guess what) math.mit.edu )