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Regularization Of Moduli Spaces Of Pseudholomorphic Curves – Katrin Wehrheim

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  • A more detailed course description
  • Access to the live stream Monday/Wednesday 11-12:30 PST
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  • References

This course aims to build a knowledge base on regularization techniques for moduli spaces arising from geometric PDE’s (in particular pseudholomorphic curves). Regularization techniques of the type discussed here were first developed in the 1980s as part of the construction of Gromov-Witten invariants (from pseudoholomorphic curves) and Donaldson invariants (from Yang-Mills instantons). They have since seen rapid extensions in the construction of a wealth of invariants for symplectic or contact manifolds as well as low dimensional manifolds, such as various Floer theories, Fukaya’s A-infty category, symplectic field theory, and gauge theoretic TQFTs. Due to the speed of the development of these applications, the subtleties of the underlying regularization techniques have not been studied systematically yet.


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
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
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Feb 5 Construction of the gluing map

  • Newton iteration
  • analytic details in (transverse) Hamiltonian Floer theory
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Feb 10 Gluing in Hamiltonian Floer theory

  • construction of the gluing map
  • topological properties of the gluing map
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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
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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
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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
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Mar 5 Isotropy and Groupoids

  • Stabilizer/isotropy groups of pseudoholomorphic maps
  • Examples of multivalued transverse perturbations
  • Groupoid language for orbifolds
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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
video slides
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 Del\ igne-Mumford 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 McDuff-Wehrheim)
  • analytic construction of morphisms
  • algebraic structure of morphisms
  • topological challenges: Hausdorffness, compactness, auxiliary metrics
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Mar 19  Kuranishi Regularization Results – a rough literature review

  • algebraic challenges
  • topological refinement results
  • ((tbd)) analytical challenges: sums of obstruction bundles
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Mar 31  Regularization approach via Polyfold Fredholm sections

  • other notions of partially smooth Banach manifolds and generalized Fredholm sections
  • an infinite dimensional regularization theorem
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Apr 2 Core ideas of polyfold theory

  • scale calculus arising from reparametrization action
  • splicings arising from pregluing
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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
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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
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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
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 non-nodal pseudoholomorphic
  • elliptic operators as scale Fredholm operators
  • towards the implicit function theorem in scale calculus
video slides
Apr 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
video slides
Apr 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
video slides
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
video slides
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
video slides