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
Sard-Smale 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 Gromov-Witten 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
Gromov-Witten invariants
compatibility of Kuranishi structure for global Fredholm section
partial differentiablility for Cauchy-Riemann operator over nontrivial Deligne-Mumford 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 McDuff-Wehrheim)
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
youtube
L15
April 2
Morse and PSS moduli spaces
compactified spaces of infinite, half infinite, and finite Morse trajectori\
es
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 non-nodal 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 M-polyfold 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 M-polyfold chart
youtube
L22
April 30
M-polyfolds
abstract notion of M-polyfold
sc retracts and splicings
a finite dimensional example
the anti-pre-gluing splicing
pregluing as M-polyfold chart in a Morse example
youtube
L23
May 5
M-polyfold bundles and Fredholm sections
Example: M-polyfold ambient space for a Morse trajectory space in C^n
M-polyfold bundle and section given by the gradient flow equation
abstract notion of M-polyfold bundle and Fredholm section thereof
implicit function theorem for transverse M-polyfold Fredholm sections
youtube
L24
May 7
M-polyfold Regularization Theorem
Sketch of bundle splicing (resp. retraction of bundle type) for Gromov-Witten moduli spaces
Fredholm filling for the Cauchy-Riemann operator in GW-setting
Transversality for Fredholm sections and their Fredholm fillings
Regularization theorem for proper Fredholm sections of M-polyfold bundles
Sketch of proof and additional features of the abstract perturbations used for M-polyfold Regularization
Addendum: Some further notions needed to construct regularizing perturbations: Strong bundle, sc^+ section, and norm/neighbourhood controlling compactness
youtube
L25