Date 
Topics 
Video 
Notes 
Jan 22 
Introduction
basic notation from symplectic manifolds via pseudoholomorphic maps/curves to (compactified) moduli spaces
some regularization slogans
a discussion of what I did (and didn't) prove in the previous course (see piazza) regarding pseudoholomorphic spheres in the context of Gromov nonsqueezing

youtube

L0 
Jan 27 
Introduction to Regularization
a finite dimensional regularization theorem
a corollary that provides a fundamental class in the Chech homology of the unregularized space
generalizations
limitations of the regularization theorem with a view towards applying it to moduli spaces of pseudoholomorphic curves
The topic of the course could be described as the quest to find generalizations of this regularization theorem that are both true and applicable to moduli spaces of pseudoholomorphic curves.
 Sorry, not captured 
L1A 

Introduction to moduli spaces
a general form for moduli spaces of pseudoholomorphic curves
discussion how it does/doesn't fit into the general form of the regularization theorem
 
L1B 
Jan 29 
Moduli spaces and their analytic description
various examples of moduli spaces of pseudoholomorphic curves
local slices to the reparametrization action on spaces of maps, yielding local Fredholm descriptions of the {holomorphic maps modulo reparametrization} part of the moduli spaces

youtube

L2 
Feb 3 
Introduction to gluing analysis
nodal/broken curves as fiber products
Gromov topology on the ``compactified'' moduli space
pregluing

youtube

L3 
Feb 5 
Construction of the gluing map
Newton iteration
analytic details in (transverse) Hamiltonian Floer theory

youtube

L4 
Feb 10 
Gluing in Hamiltonian Floer theory
construction of the gluing map
topological properties of the gluing map

youtube

L5 
Feb 12 
Geometric Regularization at the example of Hamiltonian Floer theory
summary of analytic description of moduli spaces
local injectivity of the gluing map
construction of the compactified moduli space

youtube

L6 
Feb 19 
Geometric Regularization
general philosophy and structure of the approach
local surjectivity of the gluing map in Hamiltonian Floer theory
Abstract Regularization
general philosophy and structure of the approach
Fredholm stabilization and finite dimensional reduction

youtube

L7 
Feb 24 
Regularization Philosophies
comparison/classification of geometric and abstract regularization
examples of obstructions to the equivariant transversality required by geometric regularization
philosophical approaches to extracting Euler class / fundamental class from local Fredholm descriptions
gluing in nontransverse cases  via Fredholm stabilization

youtube

L8 
Feb 26 
Transversality in Geometric Regularization
Equivariant transversality in geometric regularization
SardSmale theorem for universal moduli space
somewhere injectivity requirements
guiding questions for studying regularization approaches

youtube

L9 
Mar 5 
Isotropy and Groupoids
Stabilizer/isotropy groups of pseudoholomorphic maps
Examples of multivalued transverse perturbations
Groupoid language for orbifolds

youtube

L10 
Mar 10 
Euler class Regularization approach  overview of Siebert's work on GromovWitten moduli spaces
topological Banach manifolds with local differentiable structure
Fredholm sections that are differentiable up to finite dimensions in local models
a stabilization procedure in this context yielding an Euler class

youtube

L11 
Mar 12 
Global Fredholm description in Euler class approach
GromovWitten invariants
compatibility of Kuranishi structure for global Fredholm section
partial differentiablility for CauchyRiemann operator over nontrivial Del\
igneMumford spaces
naive attempt at Fredholm description near nodal curves
an executive summary of regularization approach #2 via finite dimensional \
reductions

youtube

L12 
Mar 17 
Kuranishi Regularization appropach  overview of various approaches based on finite dimensional reductions
categorical formulation (by McDuffWehrheim)
analytic construction of morphisms
algebraic structure of morphisms
topological challenges: Hausdorffness, compactness, auxiliary metrics
 youtube 
L13 
Mar 19 
Kuranishi Regularization
algebraic challenges
topological challenges
refinement and regularization results
 youtube 
L14 
Mar 31 
Introduction to the Arnold Conjecture
and a sketch of proof that does not require S^1 equivariance
 youtube 
L15 
April 2 
Morse and PSS moduli spaces
compactified spaces of infinite, half infinite, and finite Morse trajectories
PSS moduli spaces
proof of the Arnold conjecture for regular J and no sphere bubbling
 youtube 
L16 
April 7 
no lecture
 

April 9 
Compactification of PSS moduli spaces
fibre product description of PSS moduli spaces
proof of the Arnold conjecture in terms of abstract regularization black boxes
 youtube 
L17 
April 14 
Regularization of PSS moduli spaces
description as fiber products of SFT moduli spaces and Morse trajectory spaces
construction of PSS maps from abstract regularization
 youtube 
L18 
April 16 
General form and properties of abstract regularization theories
Fredholm properties and index of abstract sections cutting out compact moduli spaces
general abstract regularization theorems for abstract Fredholm sections
proof of relations between PSS maps from abstract regularization with boundary
 youtube 
L19 
April 21 
Polyfold overview and Scale Calculus
the regularization theorem for polyfold Fredholm sections
scale calculus
scale smoothness of reparametrization action
 youtube 
L20 
April 23 
Scale Calculus in practice
comparison with classical calculus
scale smoothness of morphisms between Fredholm descriptions of nonnodal pseudoholomorphic curves
elliptic operators as scale Fredholm operators
towards the implicit function theorem in scale calculus
 youtube 
L21 
April 28 
Scale Fredholm theory and pregluing as Mpolyfold chart
definition and practical criteria for nonlinear scale Fredholm property
implicit function theorem for nonlinear scale Fredholm maps
Cauchy Riemann operator as scale Fredholm map
towards Fredholm description near nodal curves
formalization of pregluing as Mpolyfold chart
 youtube 
L22 
April 30 
Mpolyfolds
abstract notion of Mpolyfold
sc retracts and splicings
a finite dimensional example
the antipregluing splicing
pregluing as Mpolyfold chart in a Morse example
 youtube 
L23 
May 5 
Mpolyfold bundles and Fredholm sections
Example: Mpolyfold ambient space for a Morse trajectory space in C^n
Mpolyfold bundle and section given by the gradient flow equation
abstract notion of Mpolyfold bundle and Fredholm section thereof
implicit function theorem for transverse Mpolyfold Fredholm sections
 youtube 
L24 
May 7 
Mpolyfold Regularization Theorem
Sketch of bundle splicing (resp. retraction of bundle type) for GromovWitten moduli spaces
Fredholm filling for the CauchyRiemann operator in GWsetting
Transversality for Fredholm sections and their Fredholm fillings
Regularization theorem for proper Fredholm sections of Mpolyfold bundles
Sketch of proof and additional features of the abstract perturbations used for Mpolyfold Regularization
Addendum: Some further notions needed to construct regularizing perturbations: Strong bundle, sc^+ section, and norm/neighbourhood controlling compactness

youtube 
L25 