LECTURES - VIDEOS AND NOTES

Also see site with embedded videos, or the youtube playlist.

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