The seminar takes place on Thursdays 3:30 pm - 4:30 pm in room 2-449.
Please contact Shaoyun (shaoyunb@mit.edu) for seminar-related matters.
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Organizers: Shaoyun Bai and Paul Seidel.
| Date | Speaker | Title and Abstract |
|---|---|---|
| Sep 12 |
Sebastian Haney (Columbia) | Open enumerative mirror symmetry for lines in the mirror quintic
Show/hide abstractOne of the early achievements of mirror symmetry was the prediction of genus zero Gromov-Witten invariants for the quintic threefold in terms of period integrals on the mirror. Analogous predictions for open Gromov-Witten invariants in closed Calabi-Yau threefolds can be formulated in terms of relative period integrals on the mirror, which govern extensions of variations of Hodge structure. I will discuss work in preparation in which I construct an immersed Lagrangian in the quintic and show that it supports a family of objects in the Fukaya category mirror to vector bundles on lines in the mirror quintic. The open Gromov-Witten invariants of this Lagrangian are irrational, and can be computed from homological mirror symmetry. |
| Sep 19 | Yunpeng Niu (Stony Brook) |
A Monoidal Structure on the Fukaya Category
Show/hide abstractHomological mirror symmetry suggests that there should exist on the Fukaya category certain structures analogous to those known on the derived category of coherent sheaves. In 2010, A. Subotic associated a natural monoidal structure on the generalized Donaldson-Fukaya category, mirror to the tensor product of coherent sheaves, when the symplectic manifold is equipped with a smooth Lagrangian fibration with a distinguished Lagrangian section. In this talk, I will describe a generalization of this result: how given a Lagrangian fibration (with section) with focus-focus type singularities on a symplectic 4-fold M, one can associate a natural immersed Lagrangian correspondence from M \times M to M. In the case of a generic elliptic K3, I will explain why such a correspondence is unobstructed and how it induces a monoidal structure on the Fukaya category of M. This is part of joint work with M. Abouzaid and N. Bottman. |
| Sep 26 | Guangbo Xu (Rutgers) | Quantum Kirwan map and quantum Steenrod operation
Show/hide abstractGiven a symplectic manifold with a Hamiltonian -action, the classical Kirwan map from the equivariant cohomology to the cohomology of the symplectic reduction preserves the multiplicative structure. If we consider the equivariant quantum cohomology and the quantum cohomology of the reduction space, both defined via counting holomorphic spheres, then Salamon conjectured that a quantum deformation of the Kirwan map can be defined by counting certain gauge-theoretic objects (affine vortices). I will explain my recent proof of this conjecture under certain monotonicity condition. Moreover, if we further extend the quantum Kirwan map -equivariantly, then one can define a map which intertwines the classical Steenrod operations on classifying spaces and the quantum Steenrod operation (defined by Seidel-Wilkins) on the symplectic reduction. |
| Oct 3 | Semon Rezchikov (Princeton) | Cyclotomic Structure In Symplectic Topology
Show/hide abstractSymplectic cohomology is a fundamental invariant of a symplectic manifold M with contact type boundary that is defined in terms of dynamical information and counts of pseudoholomorphic genus zero curves, and carries algebraic structures that parallel the algebraic structures on the Hochschild (co)homology of the Fukaya category of M. We show, under natural topological assumptions, that the symplectic cohomology is the homology of a genuine p-cyclotomic spectrum in the sense of Nikolaus-Scholze. The cyclotomic structure arises geometrically from the map which sends loops in M to their p-fold iterates. The data of this refinement is expected to produce many new algebraic structures of an arithmetic nature on symplectic cohomology, analogously to the way that prismatic cohomology refines the de Rham cohomology of a variety. The talk will explain the result and, if time permits, discuss concrete connections to equivariant string topology, to equivariant Floer homology operations, and to arithmetic geometry. |
| Oct 17 | Keeley Hoek (Harvard) | Family Floer theory via Morse theoretic technology
Show/hide abstractThe family Floer program gives a modern reinterpretation of the construction of the mirror of an SYZ fibration \pi : X \to B, namely as a moduli space of objects of the Fukaya category of X supported on the fibers of the fibration. The resulting rigid analytic mirror \check{X} comes equipped with a functor from the Fukaya category of X into coherent sheaves on \check{X} which can then, as an application, be used to try to prove homological mirror symmetry. In this talk I will discuss a new Morse theoretic model of family Floer theory via pseudoholomorphic treed disks, extending ideas of Charest--Woodward and Auroux. |
| Oct 22 | Nick Sheridan (Edinburgh), in 2-361 , 4pm - 5pm | Homological mirror symmetry for Calabi-Yau hypersurfaces in toric varieties
Show/hide abstractKontsevich's homological mirror symmetry conjecture predicts an equivalence between the derived category of coherent sheaves on one algebraic variety X, and the Fukaya category of a 'mirror' symplectic manifold Y. A broad class of conjectural mirror pairs (X,Y) of Calabi-Yau hypersurfaces in toric varieties was constructed by Batyrev. We prove Kontsevich's conjecture for a large subset of such Batyrev mirror pairs, in all but finitely many characteristics. The key ingredients are Gammage-Shende's proof of homological mirror symmetry 'at large volume' (which uses Ganatra-Pardon-Shende's work to reduce the computation of the Fukaya category to microlocal sheaf theory), and a deformation argument to a neighbourhood of the large volume limit using Seidel's relative Fukaya category. Based on joint work with Sheel Ganatra, Andrew Hanlon, Jeff Hicks, and Dan Pomerleano. |
| Oct 31 | Kai Hugtenburg (Lancaster) | Open Gromov-Witten invariants and Mirror symmetry
Show/hide abstractThis talk reports on two projects. The first work (in progress), joint with Amanda Hirschi, constructs (genus 0) open Gromov-Witten invariants for any Lagrangian submanifold using a global Kuranishi chart construction. As an application we show open Gromov-Witten invariants are invariant under Lagrangian cobordisms. I will then describe how open Gromov-Witten invariants fit into mirror symmetry, which brings me to the second project: obtaining open Gromov-Witten invariants from the Fukaya category. |
| Nov 7 | Yash Deshmukh (IAS) | Plumber’s algebra structure on symplectic cohomology
Show/hide abstractI will introduce a new structure on (relative) symplectic cohomology defined in terms of a PROP called the “Plumber’s PROP.” This PROP consists of nodal Riemann surfaces, of all genera and with multiple inputs and outputs, satisfying a condition that ensures the existence of positive Floer data on the surfaces. This action is defined on the chain-level and generalizes the work of Abouzaid–Groman–Varolgunes. I will discuss the relationship of this structure to cohomological field theories, with potential applications to curve counts, as well as algebraic structures defined on variants of symplectic cohomology such as Rabinowitz Floer cohomology. |
| Nov 14 | Lea Kenigsberg (UC Davis) | The string topology coproduct and why it is so cool
Show/hide abstractLet $M$ be a smooth manifold, for example a higher dimensional sphere. The free loop space is the space of all maps from the circle to $M$. Theloop space admits a very rich structure which reveals geometric andtopological information about manifoldsIn this talk I will describe the string topology coproduct, and the kind of information it encodes. In particular, I will discuss connections to invariants from Algebraic $K$ theory and Fixed point theory. This is based on joint work with Noah Porcelli. |
| Nov 21 | Joshua Greene (Boston College) | Symplectic geometry and inscription problems
Show/hide abstractThe square peg problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the place contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb. |
| Dec 4 | Eric Kilgore (Stanford), in 2-361 , 12 - 1pm | Legendrian non-squeezing via microsheaves
Show/hide abstractIn this talk I will explain some quantitative embedding results for Legendrian submanifolds of pre-quantization spaces. To start, I will recall some contact non-squeezing results for domains, and present an elementary proof of Legendrian non-squeezing for lifts of integral Lagrangian loops in T*R, using some variations on normal rulings. Then I will explain how to generalize this to high dimensions using the language of microsheaves. If time permits, as an application, I will show that the Legendrian lifts of certain monotone Lagrangian tori in \mathbb{C}^n are not squeezable. |
| Dec 5 | Jie Min (UMass Amherst), in 2-361 | Symplectic log Calabi-Yau divisors and almost toric fibrations
Show/hide abstractLagrangian fibrations sit at the crossroads of integrable systems, toric symplectic geometry and mirror symmetry. A particularly simple and interesting class of Lagrangian fibrations is called almost toric fibrations, whose total spaces are symplectic 4-manifolds. In this talk I will introduce almost toric fibrations over disks and their boundary preimages, which are symplectic divisors representing the first Chern class, called symplectic log Calabi-Yau divisors. I will then talk about joint work with Tian-Jun Li and Shengzhen Ning, showing that given a symplectic log Calabi-Yau divisor, an almost toric fibration can be constructed. |