Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analysis of a variety of geometric elliptic PDEs, in particular, pseudoholomorphic curves. It systematically addresses the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. By fall 2018, the polyfold-Fredholm property of Gromov-Witten, SFT, and Lagrangian boundary condition moduli spaces will be established, and the abstract theory has been formulated as black box that essentially allows one to argue as if moduli spaces are the zero sets of smooth sections of finite dimensional orbibundles. So it is timely for this project to serve as hands-on training in applying the polyfold framework to prove geometric applications of pseudoholomorphic curve moduli spaces. The choice of concrete applications will be made by participants and could range from elegant short proofs of classical results (e.g. Gromov nonsqueezing) via contributing to larger infrastructure (e.g. extending the work at www.polyfolds.org on Fukaya categories) to workshopping projects arising from participants' or other projects' interests, or a rather bold goal of casting adiabatic limits (e.g. between gauge theoretic equations and pseudoholomorphic curves) into the polyfold framework -- thus extending Atiyah-Floer type results beyond monotone settings.

Preferred background: You should have some experience with the analysis or geometric applications of moduli spaces of pseudoholomorphic curves or similar geometric PDEs. While we will not (need to) study the details of polyfold theory, we will in preparation for the workshop do a (guided, collaborative) reading of parts of https://arxiv.org/pdf/1210.6670.pdf, so e.g. the beginnings of sections 2.1 and 4.1 should look accessible to you.

Description: Mirror Symmetry relates the A-model defined by the symplectic structure on a manifold to the B-model defined by the complex structure of a mirror manifold. We will study homological mirror symmetry concerning Fukaya categories and Lagrangian Floer theory, as well as more traditional mirror symmetry concerning quantum cohomology and Gromov-Witten invariants counting J-holomorphic curves.

Preferred background: Familiar with basics of either Lagrangian Floer theory or Gromov-Witten theory.

It is a classical theorem of Alexander that every link in the three-sphere can be realized as a closed braid, and Bennequin later proved a version of this theorem for transverse links with respect to the standard tight contact structure on the three-sphere. There are some explicit connections between dynamical properties of a braid conjugacy class (viewed as a mapping class of the punctured disk) and surfaces the braid closure (viewed either in its smooth or transverse isotopy class) can bound in the three-sphere and the four-ball. We will investigate questions related to this observation, using Khovanov homology and Heegaard-Floer homology. In particular, we will be interested in how geometric properties of closed braid representatives of a link are reflected in algebraic properties of these homological invariants.

Preferred background: Knot and braid theory and the basics of handlebody theory and contact geometry, as well as familiarity with homology theories for knots and links.

Weinstein manifolds are a class of exact symplectic manifolds defined by Morse theory decompositions compatible inducing contact structures on level sets. This allows us to describe, for example, Weinstein 4-manifolds using Legendrian knots in S^3. This is analogous to Kirby calculus in smooth topology but strictly enriches it with the underlying geometries. Pseudoholomorphic invariants of these knots, such as Legendrian contact homology, also correspond to invariants of the Weinstein manifold, such as the wrapped Fukaya category. Similar to the smooth story, any two decompositions of a given Weinstein manifold can be induced by Kirby moves transforming the Legendrian knots into new knots. The interactions between these Kirby moves and the corresponding Fukaya invariants have many questions still to be explored, both in the general theory as well as many interesting examples.

Preferred background: Basic symplectic topology, Lagrangian Floer homology and smooth Morse/handlebody theory.

A classic problem is to understand Legendrian submanifolds up to the equivalence given by Legendrian isotopy. Motivated by ideas from Symplectic Field Theory, a coarser relation is to understand when Legendrian submanifolds are related by Lagrangian cobordism. The classical Thurston-Bennequin and rotation number invariants of Legendrian submanifolds give obstructions to the existence of a Lagrangian cobordism. There are also non-classical obstructions coming from Legendrian contact homology defined through the techniques of holomorphic curves or generating families. This project will explore some of the many open questions about the existence and obstructions to Lagrangian cobordisms, fillings, and caps.

Preferred background: Basics of Legendrian and Lagrangian submanifolds, some familiarity with Legendrian contact homology and/or generating family homology.

Moment maps are a key tool in symplectic geometry. They are used to study global topological questions via Morse theory and to construct new symplectic manifolds from old via symplectic reduction. Toric symplectic manifolds are completely determined by their moment map image. Similarly, moment maps and symmetry data govern the behavior of complexity one spaces and of toric origami manifolds. Moment map data have also been used to resolve questions in symplectic topology. Because they provide local models for Hamiltonian T-spaces, one can use them to find lower bounds for Gromov width, for example. Similar "soft" techniques have also proved useful in computing upper bounds. In this group, we will probe questions about the global topological and symplectic invariants of these various manifolds. We expect to use some computer experimentation to formulate conjectures.

Preferred background: Working knowledge of symplectic geometry, Hamiltonian actions and momentum maps. General knowledge of classical algebraic topological invariants and/or ECH capacities. Some coding experience could be helpful (but not necessary!).