Regularization Of Moduli Spaces Of Pseudholomorphic Curves – Katrin Wehrheim

• 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

• 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

• 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

• nodal/broken curves as fiber products
• Gromov topology on the “compactified” moduli space
• pregluing

• Newton iteration
• analytic details in (transverse) Hamiltonian Floer theory

• construction of the gluing map
• topological properties of the gluing map

• summary of analytic description of moduli spaces
• local injectivity of the gluing map
• construction of the compactified moduli space

• 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

• 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

• Equivariant transversality in geometric regularization
• Sard-Smale theorem for universal moduli space
• somewhere injectivity requirements
• guiding questions for studying regularization approaches

• Stabilizer/isotropy groups of pseudoholomorphic maps
• Examples of multivalued transverse perturbations
• Groupoid language for orbifolds

• 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

• Gromov-Witten invariants
• compatibility of Kuranishi structure for global Fredholm section
• partial differentiablility for Cauchy-Riemann operator over nontrivial Del\ igne-Mumford spaces
• naive attempt at Fredholm description near nodal curves
• an executive summary of regularization approach #2 via finite dimensional \ reductions

• categorical formulation (by McDuff-Wehrheim)
• analytic construction of morphisms
• algebraic structure of morphisms
• topological challenges: Hausdorffness, compactness, auxiliary metrics

• algebraic challenges
• topological refinement results
• ((tbd)) analytical challenges: sums of obstruction bundles

• 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

• 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

• description as fiber products of SFT moduli spaces and Morse trajectory spaces
• construction of PSS maps from abstract regularization

• 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

• the regularization theorem for polyfold Fredholm sections
• scale calculus
• scale smoothness of reparametrization action

• 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

• 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

• 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

• 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

• 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