Mohammed Abouzaid (Columbia)
Title: Revisiting the Arnold conjecture
Abstract: I will discuss joint work with Blumberg leading to the following result: for every closed symplectic manifold, the number of time-1 orbits of a non-degenerate Hamiltonian is bounded below by the rank of the cohomology with coefficients in any field. The case of characteristic 0 recovers a celebrated theorem of Fukaya and Ono.
Denis Auroux (Harvard Univeristy)
Title: Fukaya categories and singularities.
Abstract: Given a singular complex variety which arises as a critical level set of a regular function on a smooth ambient space, there are (at least) two ways in which one could define its Fukaya category. One of them starts from a smoothing of the hypersurface and proceeds via categorical localization; the other one involves considering a higher-dimensional Landau-Ginzburg model. We conjecture that the two constructions yield equivalent categories, and will provide evidence for some classes of examples. We will in particular focus on the toy example of the singular curve {xy=0} in the affine plane, which already has a rich geometry and can be understood from a number of different perspectives. (Part of this talk reports on work in progress by my student Maxim Jeffs).
Paul Biran (ETH)
Title: Metric Measurements and Fukaya Categories
Abstract: We introduce new metrics on the space of Lagrangian submanifolds coming from Lagrangian cobordism and filtered Fukaya categories. If time permits we will also show how these considerations can be used to review older problems on metric structures in Lagrangian topology. Based on joint works with Octav Cornea and with Egor Shelukhin.
Kai Cieliebak (Augsburg)
Title: String topology and perturbative Chern-Simons theory
Abstract: One instance of Chas and Sullivan's "string topology" is a Lie bialgebra structure on the equivariant loop space homology of a closed oriented manifold. An attempt to construct the underlying chain level structure leads to integrals over configuration spaces closely related to those in perturbative Chern-Simons theory. After recalling the relevant theories, I will explain the difficulties with this approach due to singular integrands and hidden faces, and how they can be resolved.
Aliakbar Daemi (SCGP)
Title: Exotic Structures and Equivariant Floer Homology
Abstract: There are many 4-manifolds which admit exotic smooth structures. However, it is still an open question whether there are exotic structures on simple closed 4-manidols such as the 4-dimensional sphere (smooth Poincare conjecture) and S^1xS^3. Motivated by the latter case, Akbulut asked whether there are an integral homology sphere Y with non-trivial Rokhlin invariant and a simply connected homology cobordism from Y to itself. In this talk, I will introduce various invariants of homology cobordism classes of 3-manifolds and discuss their implications about Akbulut’s question.
Yakov Eliashberg (Stanford)
Title: Arborealization: the status report.
Abstract: A few years ago David Nadler conjectured that any Weinstein manifold admits a Lagrangian skeleton with the standard, so called arboreal singularities. It turned out that there are constraints on the topology of Weinstein manifolds with arboreal skeleta. In the talk I will discuss the results and open problems. This is a joint work with Daniel Alvarez-Gavela, David Nadler and Laura Starkston.
Mark Gross (Cambridge)
Title: Intrinsic Mirror Symmetry
Abstract: In joint work with Bernd Siebert, we propose a general construction of mirror pairs. More precisely, given a pair (Y,D) with Y a smooth projective variety and D a normal crossings divisor, in certain cases we are able to define a ring using algebro-geometric methods which should be analogous to a certain piece of symplectic homology in nice cases. This includes the log Calabi-Yau case. This ring should be the ring of functions on the mirror family. If (Y,D) is sufficiently degenerate, then this ring of functions is expected to be sufficient information to recover the mirror. This approach also applies to degenerations of Calabi-Yau varieties, permitting the construction of the homogeneous coordinate ring of the mirror.
Helmut Hofer (IAS)
Title: Feral Curves and Minimal Sets
Abstract: Theories like Symplectic Field Theory use periodic orbits to build symplectic invariants for odd-dimensional manifolds with a stable Hamiltonian structure and symplectic cobordisms between them. Having a stable Hamiltonian structure is a rather strong condition, but one knows that without it, periodic orbits might not exist, i.e. the building blocks for the theory are gone. It is, of course, hard to believe that a symplectic cobordism suddenly ceases to have any meaningful symplectic properties. In this talk we present strong evidence that there is still a lot of structure which, interestingly, is related to important dynamical questions. A new class of so-called feral (pseudoholomorphic) curves relates symplectic properties to more general closed invariant subsets. As one of the applications we answer a question raised by M. Herman during his 1998 ICM talk by showing that a compact regular Hamiltonian energy surface in ${¥mathbb R}^4$ has a proper closed invariant subset. This is joint work with Joel W. Fish.
Ko Honda (UCLA)
Title: Convex hypersurface theory in higher-dimensional contact topology
Abstract: Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.
Shouhei Honda (Tohoku)
Title: Embedding of metric measure spaces in L^2 via eigenfunctions
Abstract: We discuss embedding of metric measure spaces with Ricci bounds from below in L^2 via eigenfunctions, which generalizes the classical B¥'erard-Besson-Gallot theorem on closed manifolds to singular spaces. By combining with the compactness of the moduli space of such spaces, we get the quantitative convergence result for noncollapsed spaces, which is new even for closed manifolds. This is a joint work with L. Ambrosio, J. W. Portegies and D. Tewodrose.
Kentaro Hori (Kavli IPMU)
Title: Seiberg duality and its consequences
Abstract: Seiberg duality is a low energy equivalence between pairs of supersymmetric gauge theories in 4,3,2 dimensions with four supercharges. I will introduce this duality and some of its mathematical consequences.
Kei Irie (Univeristy of Tokyo)
Title: Equidistributed periodic orbits of $C^¥infty$-generic three-dimensional
Reeb flows
Abstract: We explain a proof of the following result: for a $C^¥infty$-generic contact form $¥lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d¥lambda$. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss some speculations on possible extensions of this result, supported by known computations of ECH capacities of toric domains.
Suguru Ishikawa (RIMS, Kyoto University)
Title: Construction of general symplectic field theory
Abstract: Recently, we succeeded in the construction of symplectic field theory. I will explain about this work. Especially, I will explain the difficulty in treating the curves of genus $\geq 1$ in a space which may split.
Dominic Joyce (Oxford)
Title: “What is a Kuranishi space?”
Abstract: Kuranishi spaces were defined by Fukaya and Ono in 1999, as the geometric structure on moduli spaces of J-holomorphic curves in symplectic geometry. They were subsequently used by Fukaya-Oh-Ohta-Ono in their work on Lagrangian Floer theory and symplectic geometry. From the beginning it was clear that the definition of Kuranishi space was not entirely satisfactory (for example, there was no notion of morphism between Kuranishi spaces), and the details of the definition changed as their work developed.
We will explain a new definition of Kuranishi space (Joyce 2014), which is based on the idea that Kuranishi spaces are derived smooth orbifolds, where “derived” is in the sense of Derived Algebraic Geometry of Jacob Lurie and Toen-Vezzosi. As DAG didn’t really get started until 2004, Fukaya-Ono Kuranishi spaces predate DAG by 5 years, so one could claim that Professor Fukaya is one of the early inventors of Derived Algebraic Geometry.
An important lesson of DAG is that “derived” objects should always live in higher categories (infinity-categories or 2-categories) rather than ordinary categories. Our Kuranishi space theory is set in the world of 2-categories. So, for instance, in defining Kuranishi spaces by an "atlas of charts" approach, we have Kuranishi neighbourhoods (objects), and coordinate changes between them (1-morphisms), and crucially, also 2-morphisms between coordinate changes, which do not appear in the Fukaya-Ono theory. We develop a theory of Kuranishi spaces which form a 2-category Kur, with very nice geometric behaviour — in some ways nicer than the ordinary category of manifolds.
Our new notion of Kuranishi space can be included in the Fukaya-Oh-Ohta-Ono theory with little extra work. For example, any Fukaya-Ono Kuranishi space can be made into a Kuranishi space in our new sense, uniquely up to equivalence in Kur. It adds new possibilities within the Fukaya-Oh-Ohta-Ono theory. For example, forgetful morphisms between moduli spaces of J-holomorphic curves which forget a marked point should be 1-morphisms of new-model Kuranishi spaces, and one should use this to arrange for relations between the virtual chains of the moduli spaces.
Janko Latschev (Hamburg)
Title: Another look at algebraic structures on symplectic homology
Abstract: I will give a status report on joint work with Kai Cieliebak on the construction of various chain level algebraic structures in Hamiltonian Floer theory, particularly in symplectic homology.
Ciprian Manolescu (UCLA)
Title: SL(2,C) Floer homologies for knots and three-manifolds
Abstract: I will explain the construction of some new homology theories for knots and three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. These invariants are models for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres and small knots in S^3, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. The three-manifold construction is joint work with Mohammed Abouzaid, and the one for knots is joint with Laurent Cote.
Mark McLean (Stony Brook)
Title: Birational Calabi-Yau manifolds have the same small quantum products
Abstract: We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of symplectic cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.
Hiraku Nakajima (Kavli IPMU)
Title: Coulomb branches of quiver gauge theories and Kac-Moody Lie algebras
Abstract: The Coulomb branch of a 3d N=4 SUSY gauge theory was mathematically defined by Braverman, Finkelberg and myself a few years ago. People have been observing that it has mysterious relations to the corresponding Higgs branch, which is a hyperKaehler quotient. I will explain a conjectural realization of the Kac-Moody Lie algebra representation on the direct sum of homology groups of lagrangian subvarieties in Coulomb branches of quiver gauge theories, as an example of the relation.
Yong-Geun Oh (IBS-CGP, Postech)
Title: A wrapped Fukaya category of knot complements and hyperbolic knots
Abstract: In this talk, I will explain a construction of wrapped Fukaya category associated to the knot complement of a closed 3-manifold and construct a knot invariant called the Knot Floer Algebra. I will also discuss the proof of a formality result entering in the description of the algebra for the case of hyperbolic knots. This is based on a joint work with Youngjin Bae and Seonhwa Kim.
John Pardon (Princeton)
Title: Cosheaves of wrapped Fukaya categories
Abstract: I will discuss recent work in progress, joint with Sheel Ganatra and Vivek Shende, aimed at establishing an equivalence between certain wrapped Fukaya categories and certain categories of sheaves. Given a Weinstein manifold, we show how to decompose it into certain "Weinstein sectors", analogous to decomposing a cotangent bundle into a union of cotangent bundles of balls. The wrapped Fukaya category is a pre-cosheaf (it pushes forward under inclusions of sectors) and we aim to prove it is a genuine cosheaf (i.e. satisfies descent). This leads to an equivalence between the wrapped Fukaya category of a Weinstein manifold and the global sections of a cosheaf of categories on its Lagrangian spine.
Paul Seidel (MIT)
Title: Quantum Steenrod operations and formal groups
Abstract: Recent work on formal automorphisms of Fukaya categories points to the possible importance of quantum Steenrod operations. I will explain these and an approach towards computing them.
Nick Sheridan (Edinburgh)
Title: The Gamma and Strominger–Yau–Zaslow conjectures: a tropical approach to periods
Abstract: We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests an explanation for how the Gamma integral structure in quantum cohomology arises from the Fukaya category, using the Strominger–Yau–Zaslow conjecture. Based on joint work with Mohammed Abouzaid, Sheel Ganatra, and Hiroshi Iritani.
Ivan Smith (Cambridge)
Title: Lagrangian cobordism and rational equivalence
Abstract: There are natural maps from the Lagrangian cobordism group of a symplectic manifold X to the Grothendieck group of its Fukaya category, and from the Grothendieck group of the derived category of coherent sheaves on an algebraic variety Y to the rationalised Chow group of algebraic cycles. When X and Y are mirror, this yields a relationship between Lagrangian cobordism and rational equivalence. We investigate this and some resulting existence and non-existence results for cobordisms, focussing on the Lagrangian torus fibres of a manifold X with an SYZ fibration, and the Chow group of zero-cycles on its rigid analytic mirror Y. This talk reports on joint work with Nick Sheridan.
Gang Tian (Beijing)
Title: Gauged linear sigma model in geometric phase
Abstract: In this talk, I will explain our recent symplectic geometric construction of the cohomological field theory associated to a gauged linear sigma model (GLSM) in geometric phase. I will also explain the relation between this cohomological field theory and Gromov-Witten invariants via adiabatic limit and counting point-like instantons. This is a joint work with Guangbo Xu.
Aleksey Zinger (Stony Brook)
Title: Lifting geometric relations in real/open Gromov-Witten theory
Abstract: Many important algebraic relations between (complex) Gromov-Witten invariants have been obtained by pulling back geometric relations from the Deligne-Mumford moduli spaces of (complex) curves to Gromov's moduli spaces of stable pseudoholomorphic maps, all of which are canonically oriented. I will discuss recent progress and potential further developments concerning geometric relations on moduli spaces of real curves and the feasibility/benefit of pulling them back to moduli spaces of stable real/open pseudoholomorphic maps by not necessarily relatively orientable forgetful morphisms. This talk will be based on completed joint work with Penka Georgieva and separate work of Xujia Chen, as well as ongoing work.