UPDATED ON 20.8.18
Tobias Ekholm: Open Gromov-Witten theory, skein modules, large N duality, and knot contact homology
Large N duality relates open Gromov-Witten invariants in the cotangent bundle of the 3-sphere with closed Gromov-Witten invariants in the resolved conifold using physics arguments. We outline a symplectic geometric proof of large N duality which generalizes open Gromov-Witten invariants to invariants with values in the framed skein module and applies symplectic neck stretching. We then describe how knot contact homology and its generalizations capture the Gromov-Witten invariants ‘from infinity’ in comparatively simple terms.
Benjamin Filippenko: Fiber products of polyfolds and the PSS morphism
Often in symplectic topology we wish to constrain moduli spaces of pseudoholomorphic curves to consist of those curves satisfying intersection conditions with submanifolds. Any fiber product of moduli spaces over evaluation maps is an example of such a constraint. The evaluation maps are usually not transverse on the moduli spaces, however they extend to the ambient polyfolds and here they are submersive. Using this transversality, we construct the constrained polyfold and the constrained sc-Fredholm section, which are then used to regularize the constrained moduli space by abstract polyfold perturbation. We explain applications to Gromov-Witten invariants and the construction of the PSS morphism for general closed symplectic manifolds, which is used in an upcoming paper (joint with Katrin Wehrheim) to prove the weak Arnold conjecture in full generality.
Dan Cristofaro-Gardiner: Refined asymptotics for the ECH spectrum
In previous work, Hutchings, Ramos and I studied the embedded contact homology (ECH) spectrum for any closed three-manifold with a contact form, and proved a "volume identity" showing that the leading order asymptotics recover the contact volume. I will explain recent joint work that sharpens this asymptotic formula by better understanding the subleading term. The main technical point needed in our work is an improvement of a key spectral flow bound in Taubes' proof of the three-dimensional Weinstein conjecture; the primary goal of my talk will be to explain the ideas that go into this improvement.
Michael Hutchings: Obstruction bundle gluing
Obstruction bundle gluing is a technique for gluing configurations in which transversality fails, but one still has the nice property that cokernel dimensions do not jump. This technique can help establish the foundations of some theories. In addition, it can be used to explicitly compute the contributions to certain counts (defined by abstract perturbations) from certain non-transverse configurations. We will illustrate this technique via the simplest nontrivial example that we know about, which arises in circle-valued Morse theory.
Michael Jemison: Lego Pieces with an application to Deligne-Mumford Spaces with Boundary 1
We will discuss Deligne-Mumford spaces with boundary and their formulation in terms of ep-groupoids as an introduction to Lego piece constructions. We will set up the analytic framework to express the space of diffeomorphisms between appropriate families of degenerating Riemann surfaces with boundary as a polyfold.
Michael Jemison: Lego Pieces with an application to Deligne-Mumford Spaces with Boundary 2
We will verify the Pre-Fredholm properties for the M-polyfold models constructed in Talk 1 for Deligne-Mumford spaces. We will show that the space of biholomorphisms between families of degenerating families of Riemann surfaces with boundary is defined as the solution set of an sc-Fredholm operator. We will see that the use of Lego pieces simplifies the analysis.
Dusa McDuff: Branched Orbifold models of Polyfold Fredholm sections
I will describe the construction of a finite dimensional model of a polyfold Fredholm section that simplifies the process of finding perturbation multisections. In fact one can construct the virtual moduli cycle on the chain level in a homological way, without using such a perturbation. This construction can be made compatible with coherent systems of polyfolds. Much of this work is joint with Katrin Wehrheim and Ben Filippenko.
Wolfgang Schmaltz: Gromov-Witten Axioms for Symplectic Manifolds via Polyfold Theory
In 1994 Kontsevich and Manin stated the Gromov-Witten axioms, given as a list of formal relations between the Gromov-Witten invariants. In this talk I will prove several of the Gromov-Witten axioms for curves of arbitrary genus for all closed symplectic manifolds.
Jake Solomon: Inductive extension of multisections
I will begin by explaining why the problem of extending a smooth multivalued function from the boundary of a manifold with corners to its interior presents significant difficulties beyond both the single valued case and the case of a manifold with boundary. Then, I will present an extension construction tailored to the inductive arguments used in SFT and open Gromov Witten theory. This is joint work with H. Hofer.
Katrin Wehrheim: Family polyfold theory and adiabatic limits
I will discuss work in progress with Nate Bottman on describing adiabatic limits - such as the strip shrinking in pseudoholomorphic quilts - in terms of a single scale-smooth Fredholm section. This will require a generalization of scale calculus to vector spaces equipped with a "family of scale-structures''.
Zhengyi Zhou: Quotients of polyfolds, equivariant fundamental class and localization.
In the appearance of a group symmetry, symplectic invariants like Gromov-Witten theory and Floer theory can often be enriched to equivariant theories. Usually, the group acts on the underlying polyfolds. In this talk, we first discuss the quotients of polyfold bundles and sc-Fredholm sections, when the group action only has finite isotropy. When the base polyfold has no boundary, we construct an equivariant fundamental class for any group action preserving orientation. Combining with the Gromov-Witten polyfolds, the equivariant fundamental class defines an equivariant Gromov-Witten theory. We will then discuss the localization formula for the equivariant fundamental class.