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.

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.

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.

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.

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.

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.