The SGGTC seminar meets on Fridays in Math 520 from 10:30-11:30am and in Math 417 from 1-2pm, unless noted otherwise (in red).

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 Date Speaker Title Jan. 31, 10:30am Boyu Zhang (Princeton) Classification of n-component links with Khovanov homology of rank 2^n Feb. 7, 10:30am CANCELED: Yongbin Ruan (Zhejiang University) BCOV axioms of Gromov-Witten theory of Calabi-Yau 3-fold Feb. 7, 1pm Chris Woodward (Rutgers) Tropical Fukaya algebras Feb. 14, 10:30am Johannes Horn (Heidelberg) Singular fibers of Hitchin integrable systems Feb. 21, 1pm Daniel Álvarez-Gavela (IAS / Princeton) K-theoretic invariants for Legendrians via parametrized Morse theory Feb. 28, 10:30am Laurent Côté (Stanford) Contact homology twisted by a codimension 2 submanifold Feb. 28, 1pm Zhenkun Li (MIT) Decomposing sutured Instanton Floer homology Mar. 6, 10:30am Xin Jin (Boston College) Homological mirror symmetry for the universal centralizers Mar. 6, 1pm Søren Galatius (Copenhagen) Symplectic K-theory of the integers, and an action of Aut(C) Mar. 13, 10:30am CANCELED: Michael Sullivan (UMass Amherst) The persistence of the Legendrian contact homology algebra Mar. 13, 1pm Florian Naef (MIT) String topology and the configuration space of two points Mar. 20 Spring recess Mar. 27, 10:30am Piotr Suwara (MIT) TBA Mar. 27, 1pm Arik Wilbert (University of Georgia) TBA Apr. 3, 10:30am Jo Nelson (Rice University) TBA Apr. 3, 1pm TBA TBA Apr. 10, 10:30am Marcy Robertson (University of Melbourne) TBA Apr. 10, 1pm Jack Smith (Cambridge) TBA Apr. 17, 10:30am Hiro Lee Tanaka (Texas State University) TBA Apr. 17, 1pm Sherry Gong (UCLA) TBA Apr. 24, 10:30am Andy Manion (USC) TBA Apr. 24, 1pm Lisa Piccirillo (Brandeis / MIT) TBA May. 1, 10:30am TBA TBA May. 1, 1pm TBA TBA

# Abstracts

#### Jan 31st, 2020: Boyu Zhang (Princeton) " Classification of n-component links with Khovanov homology of rank 2^n "

Abstract: Suppose L is a link with n components and the rank of Kh(L;Z/2) is 2^n, we show that L can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and generalizes the unlink detection theorem of Khovanov homology by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology introduced by Kronheimer and Mrowka. This is joint work with Yi Xie.

#### Feb 7th, 2020: Yongbin Ruan (Zhejiang University) " BCOV axioms of Gromov-Witten theory of Calabi-Yau 3-fold "

Abstract: One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-fold such as the quintic 3-folds. There have been a collection of remarkable conjectures from physics (BCOV B-model) regarding the universal structure or axioms of higher genus Gromov-Witten theory of Calabi-Yau 3-folds. In the talk, I will first explain 4 BCOV axioms explicitly for the quintic 3-folds. Then, I will outline a solution for 3+1/2 of them.

This talk is based on the joint works with F. Janda and S. Guo and can be thought as the pre-talk to F. Janda's talk in algebraic geometry seminar at the same day.

#### Feb 7th, 2020: Chris Woodward (Rutgers) " Tropical Fukaya algebras "

Abstract: In many cases, the hardest part of computing Lagrangian Floer cohomology is to show that a particular Lagrangian is weakly unobstructed. For example, weak unobstructedness of fibers of SYZ fibrations are covered in several recent works. I will introduce a new tool, joint with Sushmita Venugopalan (Chennai) called the tropical Fukaya algebra, which is homotopy equivalent to the original Fukaya algebra and whose structure coefficients are counts of tropical graphs. The construction is similar to that of Eleny Ionel and Brett Parker, but we use a further Fulton-Sturmfels-type degeneration of the diagonal to remove all matching conditions. A corollary is that any tropical toric moment fiber is weakly unobstructed, where a tropical toric moment fiber is a fiber of a normal-crossing toric degeneration whose limit is contained in a toric component.

#### Feb 14th, 2020: Johannes Horn (Heidelberg) " Singular fibers of Hitchin integrable systems "

Abstract: Hitchin systems are an important class of algebraically completely integrable systems defined on moduli spaces of Higgs bundles. We will introduce a new approach to study singular fibers of theses systems by semi-abelian spectral data. This stratifies the singular Hitchin fibers by bundles over maximal abelian subvarieties with fibers given by parameters of higher Hecke transformations. In the talk we will concentrate on the SL(2,C)-case and explain how this leads to a complete description of a particular class of singular fibers.

#### Feb 21st, 2020: Daniel Álvarez-Gavela (IAS / Princeton) " K-theoretic invariants for Legendrians via parametrized Morse theory "

Abstract: Joint work with K. Igusa. It was remarked by Eliashberg and Gromov that any Legendrian link in the standard contact R^3 which is generated by a 1 parametric family of functions corresponding to a pseudo-isotopy with nontrivial Wh_2 invariant is not Legendrian isotopic to an unlink of standard Legendrian unknots. Similarly, one can show that any Legendrian link in R^5 which is generated by a 2 parametric family of functions corresponding to a loop of pseudo-isotopies with nontrivial Wh_3 invariant is not Legendrian isotopic to an unlink of standard Legendrian unknots. In a similar spirit, we study a family of Legendrian submanifolds in the 1-jet space of any oriented surface which carry nontrivial and distinct K_3 invariants. In our setting the product K_1 x K_2 \to K_3 together with a standardization of the K_2 factor allows us to understand these examples more simply in terms of a K_1 type invariant. We call this invariant the Legendrian Turaev torsion. As an application, we show that in the 1-jet space of any oriented surface there exist pairs of Legendrian links which (a) are formally isotopic (b) cannot be distinguished by any natural Legendrian invariant (c) yet are not Legendrian isotopic. These examples appeared in a different guise in work of K. Igusa and J. Klein on the higher Reidemeister torsion of circle bundles, where in particular an explicit picture for the exotic element of K_3(Z)=Z/48 was given.

#### Feb 28th, 2020: Laurent Côté (Stanford) " Contact homology twisted by a codimension 2 submanifold "

Abstract: There is a rich theory of transverse knots in 3-dimensional contact manifolds. It was a major open question in contact topology whether non-trivial transverse knots (i.e. codimension 2 contact embeddings) also exist in higher dimensions. This question was recently settled in the affirmative by Casals and Etnyre. Motivated by their result, I will talk about work in progress with Francois-Simon Fauteux-Chapleau to develop invariants of codimension 2 contact embeddings using the machinery of symplectic field theory.

#### Feb 28th, 2020: Zhenkun Li (MIT) " Decomposing sutured Instanton Floer homology "

Abstract: Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka. Though it has many important applications to the study of 3-dimensional topology, many basic aspects of the theory remain undeveloped. In this talk I will explain how to decompose sutured Instanton Floer homology with respect to properly embedded surfaces inside the sutured manifold, and present some applications of this decomposition to the development of the theory: performing some computations, bounding the depth of taut sutured manifolds, detecting the Thurston norm on link complements, and constructing some invariants for knots and links. The work is partially joint with Sudipta Ghosh.

#### Mar 6th, 2020: Xin Jin (Boston College) " Homological mirror symmetry for the universal centralizers "

Abstract: I will present work in progress on the homological mirror symmetry for the universal centralizer J_G associated to a complex semisimple Lie group G. The A-side is a partially wrapped Fukaya category on J_G, and the B-side is the category of coherent sheaves on the categorical quotient of the dual maximal torus by the Weyl group action (with some modification if G is not of adjoint type).

#### Mar 6th, 2020: Søren Galatius (Copenhagen) " Symplectic K-theory of the integers, and an action of Aut(C) "

Abstract: The integral symplectic group $Sp_{2g}(Z)$ is the subgroup of $GL_{2g}(\mathbb{Z})$ consisting of automorphisms of Z^{2g} which preserve the standard symplectic form. I will recall the definitions of algebraic K-theory $K_*(\mathbb{Z})$ and symplectic K-theory $KSp_*(\mathbb{Z})$ from these matrix groups, and review what is known about these important invariants. I will then explain a natural action of the group of automorphisms of the field of complex numbers on $KSp_*(Z)$ after completing at a prime p. The main result is a characterization of this action by a universal property. Joint work with Tony Feng and Akshay Venkatesh.

#### Mar 13th, 2020: Michael Sullivan (UMass Amherst) " The persistence of the Legendrian contact homology algebra "

Abstract: The well-studied displacement energy of a Lagrangian submanifold is the minimum Hamiltonian oscillation needed so that the Lagrangian does not intersect its image under the Hamiltonian isotopy. For dimension reasons, the analogous Legendrian displacement energy definition requires the contact Hamiltonian image of the Legendrian to not intersect the Reeb flow of the original Legendrian. In this contact setting, I will discuss how to apply the persistent homology barcodes of a filtered Legendrian contact (Chekanov-Eliashberg) algebra to get lower bounds on the number of such intersections. The numerical bounds vary with the size of the Hamiltonian oscillation and hold when a Legendrian has a large loose chart, in contrast to a Legendrian with a small loose chart which sometimes has a small displacement energy. I will also discuss a related Legendrian non-squeezing result. This is joint work with Georgios Dimitroglou Rizell.

#### Mar 13th, 2020: Florian Naef (MIT) " String topology and the configuration space of two points "

Abstract: Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points. This is joint work with Thomas Willwacher

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the "Floer Homology" e-mail list maintained via Google Groups.