Links to Videos, Papers, and Ongoing Work on Polyfold Theory

Videos of talks

• 2015 Summer School on Moduli Problems in Symplectic Geometry playlist [1], in particular series by J.Fish, K.Wehrheim [2], [3], [4], [5], [6]; discussions with N.Bottman [7], [8]; H.Hofer on construction of SFT polyfolds [9], [10], [11], [12], [13]

• Lecture course [Regularization Of Moduli Spaces Of Pseudoholomorphic Curves] (K.Wehrheim, 2014)

• Introduction to Polyfolds (K.Wehrheim, 2012 at IAS) [14]

• Lecture series on Polyfolds (H.Hofer, 2012 at IAS) [15], [16], [17], [18], [19], [20], [21], [22]

• An M-polyfold relevant to Morse theory (P.Albers, 2012 at IAS) [23]

• Transversality questions and polyfold structures for holomorphic disks (K.Wehrheim, 2009 at MSRI) [24]

Surveys and Textbooks

• A Polyfold Cheat Sheet (K.Wehrheim, 2016) [25]

• Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds (H.Hofer, K.Wysocki, E.Zehnder, 2014) [26]

• Polyfolds And A General Fredholm Theory (H.Hofer, 2008&2014) [27]

• Polyfolds: A First and Second Look (O.Fabert, J.Fish, R.Golovko, K.Wehrheim, 2012) [28]

• A General Fredholm Theory and Applications (H.Hofer, 2005) [29]

Abstract Polyfold Theory

• A General Fredholm Theory I: A Splicing-Based Differential Geometry (H.Hofer, K.Wysocki, E.Zehnder, 2006) [30]

• A General Fredholm Theory II: https://arxiv.org/abs/0705.1310 (H.Hofer, K.Wysocki, E.Zehnder, 2007) [31]

• A General Fredholm Theory III: Fredholm Functors and Polyfolds (H.Hofer, K.Wysocki, E.Zehnder, 2008) [32]

• Integration Theory for Zero Sets of Polyfold Fredholm Sections (H.Hofer, K.Wysocki, E.Zehnder, 2007) [33]

• Sc-Smoothness, Retractions and New Models for Smooth Spaces (H.Hofer, K.Wysocki, E.Zehnder, 2010) [34]

• Coherent M-polyfold Theory (B.Filippenko) - provides a construction scheme for perturbations of Fredholm sections (with trivial isotropy) whose boundary stratifications are Cartesian products of other Fredholm sections, while preserving transversality and compactness; preliminary draft available

• Sliceable Polyfolds and Constrained Moduli Problem (B.Filippenko) - establishes an implicit function theorem for non-Fredholm submersions cutting out 'slices' of finite codimension, proves that restrictions of polyfold Fredholm sections to 'slices' remain Fredholm, and in particular proves that suitably transverse fiber products of Polyfold Fredholm sections are again Polyfold Fredholm; preliminary draft available

• Quotients in Polyfold Theory (Z.Zhou) - studies Polyfold Fredholm sections that are equivariant under the action of a compact Lie group, constructs equivariant transverse perturbations for actions with finite isotropy and $S^{1}$ actions with vanishing obstructions, and applies this to prove Arnold conjecture; preliminary draft available

• Equivariant Fundamental Class and Localization Theorem (Z.Zhou) - constructs an equivariant fundamental class for any equivariant polyfold Fredholm section (without boundary), and proves the localization theorem for 'regular fixed point locus'; preliminary draft available

Applications of Polyfold Theory

• Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory (H.Hofer, K.Wysocki, E.Zehnder, 2011) [35]

• Fredholm notions in scale calculus and Hamiltonian Floer theory (K.Wehrheim, 2012&2016) [36]

• A-infty structures from Morse trees with pseudoholomorphic disks (Jiayong Li, K.Wehrheim, 2014 preliminary draft) [37]

• Applications of Polyfold Theory II: The Polyfolds of SFT (J.Fish, H.Hofer) - constructs Polyfold Fredholm sections whose zero sets are the SFT moduli spaces

• SFT via Polyfold Theory (J.Fish, H.Hofer) - constructs the SFT algebra package via coherent perturbations and a study of homotopies of data

• A proof of the Arnold Conjuecture by Polyfold Techniques (P.Albers, B.Filippenko, J.Fish, K.Wehrheim) - constructs the PSS isomorphism in general compact symplectic manifolds - preliminary draft and [slides] available

• Gromov-Witten axioms for polyfold construction (W.Schmaltz, K.Wehrheim) - proves the Kontsevich-Manin axioms for the polyfold construction of Gromov-Witten invariants; writing in progress

• Polyflow Homotopy Theory (Z.Zhou) - develops homotopical invariants from systems of Polyfold Fredholm sections whose zero sets- if cut out transversely - form a Cohen-Jones-Segal 'flow category', applies in Morse-Bott settings, and provides new approaches to proving invariance of algebraic structures under homotopies of data; work in progress