Regularization of moduli spaces of pseudholomorphic curves



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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 Del\ igne-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 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 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