Abstracts
When available, notes from the lectures are available by clicking on the titles. Special thanks to our note takers: Robert Castellano, Zhechi Cheng, Chris Gerig, Ziva Myer, Kathrin Naef, Samantha Pezzimenti, Kevin Sackel, and Jingyu Zhao.
Strom Borman (Stanford), Quantum cohomology via symplectic cohomology.
Abstract. Consider a monotone closed symplectic manifold M together with an ample normal crossings divisor D such that M∖D is a Liouville manifold. In this talk I will describe how in some cases the quantum cohomology of M is recovered by deforming the symplectic cohomology of M∖D via a Maurer-Cartan element in the L∞ structure of the symplectic cohomology of M∖D. This is joint work in progress with Nick Sheridan.
Olguta Buse (IUPUI), Contact embeddings and packings in dimension three.
Abstract. In joint work with D. Gay, we introduce the concepts of capacity and shape for a three dimensional contact manifold (M, ξ) relative to a transversal knot K. We will explain the connection with the existing literature (Eliashberg's concept of shape introduced in the 90's ) and provide our main computation for the shape in the case of lens spaces L(p,q). The main tools used here are rational surgeries which will be explained through their toric interpretations based on the continuous fraction expansions of p/q. We will discuss possible parallels with the study of ellipsoid embeddings in four dimensions.
Roger Casals (MIT), The Lefschetz-Front Dictionary II: Applications.
Abstract. In this series of two talks we discuss Weinstein structures endowed with Lefschetz fibrations in terms of Legendrian front projections. This second talk is focused on applications of this Lefschetz-Front dictionary. These include the description of Stein handlebodies of affine algebraic varieties, the detection of flexible and subflexible Weinstein structures and the existence of exact Lagrangian submanifolds. In particular, we will explain the role of the Clifford torus in the square of a Dehn twist and compute the mirror functor for the complement of a conic. This is joint work with Emmy Murphy.
Emily Clader (ETH Zurich), Sigma models and phase transitions.
Abstract. The Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence is a proposed equivalence between two enumerative theories associated to a homogeneous polynomial: the Gromov-Witten theory of the hypersurface cut out by the polynomial in projective space, and the Landau-Ginzburg theory of the polynomial when viewed as a singularity. Such a correspondence was originally suggested by Witten in 1993 as part of a far-reaching conjecture relating the “gauged linear sigma models” arising at different phases of a GIT quotient. I will discuss an explicit formulation and proof of Witten's proposal for complete intersections in projective space, generalizing the LG/CY correspondence for hypersurfaces and introducing a number of new features. This represents joint work with Dustin Ross.
Daniel Cristofaro-Gardiner (Harvard), Symplectic embeddings and the Fibonacci numbers.
Abstract. It can be subtle to determine whether or not one symplectic manifold can be embedded into another, even for simple domains in Cn. For example, McDuff and Schlenk computed when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that if the ellipsoid is close to round, the answer is given by an “infinite staircase” determined by the odd-index Fibonacci numbers and the Golden Mean. I will explain recent joint work with Richard Hind, showing that a version of this result holds in all dimensions.
Maia Fraser (Ottawa), Contact non-squeezing via contact homology.
Abstract. I will discuss non-squeezing in the contact manifold R2n×S1 and a proof that “large-scale” prequantized balls, namely B2n(R)×S1 with R>1, cannot be squeezed into themselves by a compactly supported contactomorphism. This statement and the proof by contact homology that I will sketch are in the same spirit as Eliashberg-Kim-Polterovich's (2006) proof of non-squeezing for R ∈ N. By contrast Chiu (2014) proved, using microlocal sheaf theory, that B2n(R)×S1, R>1 cannot be squeezed into itself by a compactly supported contact isotopy; a proof of that statement using generating functions is also possible. It is not known if this formally weaker statement is in fact equivalent to the one we prove.
Eduardo Gonzalez, (U Mass Boston), Quantum Kirwan for quantum K-theory.
Abstract. We extend, under certain assumptions, the definition of Kirwan map to quantum K-theory. This is a map from the equivariant quantum K-theory of a polarised G-variety (Hamiltonian manifold) to the quantum K-theory of the geometric invariant theory (symplectic) quotient. As an application we give a presentation of the quantum K-theory of projective spaces, and Grassmannians. This is joint work with C. Woodward.
Kristen Hendricks (UCLA), A flexible construction of equivariant Floer cohomology.
Abstract. In the past few years, equivariant Floer cohomology has been used to construct many spectral sequences between Floer-type invariants of three-manifolds and knots. We will give an alternative formulation of equivariant Lagrangian Floer cohomology, which can be used to show several of these spectral sequences are invariants of their topological input data and/or explicitly computable, and can also be applied to define new equivariant versions of several other Floer-type invariants. As an application, we construct a new knot concordance invariant which appears to be distinct from other concordance invariants from Floer theory. This is joint work with R. Lipshitz and S. Sarkar.
Michael Hutchings (Berkeley), Cube capacities.
Abstract. We study the “cube capacity" of a symplectic manifold, defined to be the size of the largest cube (symplectic product of 2-disks of equal areas) that can be symplectically embedded into it. We show that in many cases, including all convex or concave toric domains in any number of dimensions, the obvious inclusion of a cube is optimal. The proof uses new symplectic capacities, conjecturally generalizing the Ekeland-Hofer capacities, defined using S^1-equivariant symplectic homology. This is joint work with Jean Gutt.
Mark McLean (Stonybrook), The cohomological McKay correspondence and symplectic cohomology.
Abstract. Suppose that we have a finite quotient singularity Cn/G admitting a crepant resolution Y (i.e. a resolution with c1 = 0). The cohomological McKay correspondence says that the cohomology of Y has a basis given by irreducible representations of G (or conjugacy classes of G). Such a result was proven by Batyrev when the coefficient field F of the cohomology group is Q. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of Y in two different ways. This proof also extends the result to all fields F whose characteristic does not divide |G| and it gives us the corresponding basis of conjugacy classes in H*(Y). We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.
Emmy Murphy (MIT), The Lefschetz-Front Dictionary I: Construction.
Abstract. In this series of two talks we describe Weinstein structures endowed with Lefschetz fibrations in terms of Legendrian fronts. This first talk is focused on the construction and proof of this Lefschetz-Front dictionary, translating Weinstein Lefschetz fibrations to Legendrian links; the main focus is on Ak-biLefschetz fibrations. In particular, the front presentation of a Dehn twist and Legendrian handle slides will be discussed, along with the basic features of higher-dimensional Legendrian calculus. This is joint work with Roger Casals.
John Pardon (Stanford / Clay Math), Existence of Lefschetz fibrations on Stein/Weinstein domains.
Abstract. I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.
Sobhan Seyfaddini (MIT), C0 Hamiltonian dynamics and the Arnold conjecture.
Abstract. After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a C0 counter example to the Arnold conjecture in dimensions higher than two. This is joint work with Lev Buhovsky and Vincent Humilière.
Laura Starkston (Stanford), Symplectic lines, concave divisor caps, and symplectic fillings.
Abstract. We will discuss contact manifolds which arise on the boundary of certain plumbings with concave boundary. Some of these are links of singularities. By using a celebrated theorem of McDuff, the fillings of these contact manifolds can be understood in terms of embeddings of pseudoholomorphic curves into a rational or ruled surface. An interesting object which arises in such embedding problems the notion of a symplectic line arrangement. We will discuss some similarities and differences between symplectic, topological, and complex algebraic line arrangements and the implications for symplectic fillings (joint with Danny Ruberman).
Margaret Symington (Mercer), Gluing Hamiltonian integrable systems with S1-symmetry.
Abstract. In this talk I will describe an approach to gluing completely integrable Hamiltonian systems of two-degrees of freedom in which one of the two integrals is the moment map for a Hamiltonian S1-action. As a specific example, I will describe a blow-up in the semi-toric category, where the moment map for the Hamiltonian S1-action is proper and any singular points are almost-toric (elliptic-regular, elliptic-elliptic, or focus-focus). I will also explain how to use that semi-toric blow-up to prove that the conditions necessary for a Hamiltonian S1-space to underlie a semi-toric system are in fact sufficient. This is joint work with Hohloch, Sabatini and Sepe.
Susan Tolman (UIUC), Non-Hamiltonian symplectic circle actions with isolated fixed points.
Abstract. We construct a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed, connected, six-dimensional symplectic manifold. This answers a question posed by McDuff and Salamon. Based in part on joint work with J. Watts.
Michael Usher (UGA), Barcodes and chain-level Floer theory.
Abstract. As was recently exploited in Polterovich and Shelukhin's work on autonomous Hamiltonian diffeomorphisms, filtered Floer-theoretic invariants on a monotone symplectic manifold can be neatly described by the “barcodes” that have been in use for some time in the persistent homology literature. I will discuss joint work with Jun Zhang which generalizes these barcodes to the non-monotone setting. The result is a relatively-simple-to-understand invariant which completely characterizes the chain isomorphism type of a Floer-type chain complex over a field, which satisfies nice continuity properties, and in which every generator of the complex (e.g. periodic orbit or Lagrangian intersection) plays an identifiable role. I will also mention some applications to Hofer geometry, mostly coming from recent work of Zhang and Stevenson.
Katrin Wehrheim (Berkeley), Polyfold lab report.
Abstract. I will survey various results (in progress) on foundations and applications of Hofer-Wysocki-Zehnder's polyfold theory:
- Fiber products of polyfold Fredholm sections
- Equivariant transversality - existence and obstructions
- Equivariant fundamental class
- Gromov-Witten axioms
- Two polyfold proofs of the Arnold conjecture
These are joint with or due to Peter Albers, Ben Filippenko, Joel Fish, Jiayong Li, Wolfgang Schmaltz, and Zhengyi Zhou.