Date 
Topics 
Videos 
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

video 
slides 
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.
Introduction to Regularization
 a general form for moduli spaces of pseudoholomorphic curves
 discussion how it does/doesn’t fit into the general form of the regularization theorem

No Video 
slides1
slides2 
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

video 
slides 
Feb 3 
Introduction to gluing analysis
 nodal/broken curves as fiber products
 Gromov topology on the “compactified” moduli space
 pregluing

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

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

video 
slides 
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

video 
slides 
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

video 
slides 
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

video 
slides 
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

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

video 
slides 
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

video 
slides 
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

video 
slides 
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

video 
slides 
Mar 19 
Kuranishi Regularization Results – a rough literature review
 algebraic challenges
 topological refinement results
 ((tbd)) analytical challenges: sums of obstruction bundles

video 
slides 
Mar 31 
Regularization approach via Polyfold Fredholm sections
 other notions of partially smooth Banach manifolds and generalized Fredholm sections
 an infinite dimensional regularization theorem

video 
slides 
Apr 2 
Core ideas of polyfold theory
 scale calculus arising from reparametrization action
 splicings arising from pregluing

video 
slides 
Apr 7 
No Lecture 


Apr 9 
Core ideas of polyfold theory
 other notions of partially smooth Banach manifolds and generalized Fredholm sections
 an infinite dimensional regularization theorem
 scale calculus arising from reparametrization action
 splicings arising from pregluing

video 
slides 
Apr 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

video 
slides 
Apr 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

video 

Apr 21 
Polyfold overview and Scale Calculus
 the regularization theorem for polyfold Fredholm sections
 scale calculus
 scale smoothness of reparametrization action

video 
slides 
Apr 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

video 
slides 
Apr 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

video 
slides 
Apr 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

video 
slides 
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

video 
slides 
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

video 
slides 