The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).

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 Date Speaker Title Sep. 11 10:15 Math 520 Amitai Zernik (Einstein Institute of Mathematics, Hebrew University of Jerusalem) Fixed-point expressions for the Fukaya endomorphism algebra of $\mathbb{RP}^{2m}$ and higher genus open invariants 1pm Math 407 Netanel Blaier (MIT) The Quantum Johnson homomorphism, Formality and symplectic isotopy 2pm Math 520 Organizational Meeting Joint with Geometric Topology Seminar Sep. 18 Steven Sivek (Princeton University) TBA Sep. 25 Adam Levine (Princeton University) Khovanov homology and knot Floer homology for pointed links Oct 2 2pm IAS Simonyi Hall 101 Octav Cornea (University of Montreal) Lagrangian cobordism: what we know and what is it good for (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) 2pm IAS Simonyi Hall 101 Egor Shelukhin (IAS) Non-trivial Hamiltonian fibrations via K-theory quantization (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Oct. 9 Mohammad Tehrani (Simons Center, Stony Brook) Normal crossings divisors and configurations for symplectic topology Oct. 16 Adam Lowrance (Vassar College) Khovanov homology, chromatic homology, and torsion Oct. 23 Francesco Lin (MIT) Pin(2)-monopole Floer homology 2pm Math 622 Yoel Groman (ETH Z\"urich) Floer theory on open manifolds Oct. 30 10:15 Math 520 Sheila (Margherita) Sandon (Université de Strasbourg) Floer homology for translated points (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) 1pm Math 407 Sheel Ganatra (Stanford University) Functors and relations from Fukaya categories of LG models (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Nov. 6 10:30 Math 312 Vinoth Nandakumar (University of Utah) Modular representations with two-block nilpotent central character 1pm Math 407 Matt Hogancamp (Indiana University Bloomington) Categorical diagonalization of the full twist Nov. 12 11:40 Math 407 Paul Wedrich (University of Cambridge) Deformations of type A link homologies Nov. 13 Dmitry Tonkonog (University of Cambridge) Circle actions and the Fukaya category Nov. 20 Soren Galatius (Stanford University) Tautological rings for high-dimensional manifolds Nov. 27 No seminar -- Thanksgiving holiday Dec. 3 11:40 Math 507 Krzysztof Putyra (ETH-ITS) On categorical traces and homology for links in a solid torus Dec. 4 Kyler Siegel (Stanford University) Subflexible symplectic manifolds

# Abstracts

#### September 11, 2015: Amitai Zernik "Fixed-point expressions for the Fukaya endomorphism algebra of $\mathbb{RP}^{2m}$ and higher genus open invariants"

Abstract: The Atiyah-Bott localization formula has proven a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps. In contrast, the ''open'' moduli spaces, of stable maps of marked Riemann surfaces with boundary, have boundaries, and these boundaries must be taken into account in order to apply the localization formula. In this talk we'll see that (an equivariant extension of) the Fukaya $A_{\infty}$ algebra structure allows to write down expressions which effectively eliminate the boundaries in genus zero, so one can define equivariant invariants and compute them using localization. These invariants specialize to the open Gromov-Witten invariants, and in particular produce new combinatorial expressions for Welschinger's signed counts of real rational plane curves in terms of summation over certain even-odd diagrams. Time permitting, we'll discuss the two-sided information flow with the intersection theory of Riemann surfaces with boundary (mapping to a point), which lends evidence to a conjectural generalization of these formulas to higher genus. Most of this is joint work, in preparation, with Jake Solomon.

#### September 11, 2015: Netanel Blaier "The Quantum Johnson homomorphism, Formality and symplectic isotopy"

Abstract: Many problems in symplectic topology can be phrased as questions about the topology of the symplectomorphism group. We consider the problem of identifying the symplectic isotopy class of a symplectomorphism $\phi : X \to X$ which acts trivially on cohomology. When $X = \Sigma_g$ is a surface, this is the well-known Torelli group, a major object of study in low-dimensional topology with many interesting connections to other areas of mathematics. In the early 1980's, Dennis Johnson revolutionized the study of this group by introducing a sequence of homomorphisms $\tau_k$ detecting delicate intersection-theoretic information. We rephrase $\tau_k$ as measuring the non-formality of a certain $A_\infty$-algebra, and present an analogue of $\tau_2$ in symplectic topology using Quantum matrix Massey products. As a result, we show that the monodromy of a certain family of 3-folds is not isotopic to the identity.

#### September 25, 2015: Adam Levine "Khovanov homology and knot Floer homology for pointed links"

Abstract: There are spectral sequences relating reduced Khovanov homology to a variety of other homological link invariants, including the Heegaard Floer homology of the branched double cover and instanton knot homology. However, there is no known relationship between Khovanov homology and knot Floer homology, despite considerable computational evidence and numerous formal similarities. I will describe ongoing efforts to find a spectral sequence relating these two invariants. Specifically, we construct a variant of Khovanov homology for links with one or more basepoints on each component, which more closely parallels the behavior of knot Floer homology and which conjecturally fits into a spectral sequence as required.

#### October 2, 2015: Octav Cornea "Lagrangian cobordism: what we know and what is it good for"

Abstract: I will describe how the notion of Lagrangian cobordism, introduced by Arnold in 1980, offers a systematic perspective on the study of Lagrangian topology. There are three aspects that will be emphasized: the relations with the triangulated structure of the derived Fukaya category, the connection to Seidel’s representation, cobordism based extensions of the Hofer norm. The talk is based mainly on joint work with Paul Biran (ETH).

#### October 2, 2015: Egor Shelukhin "Non-trivial Hamiltonian fibrations via K-theory quantization"

Abstract: We produce examples of non-trivial Hamiltonian fibrations that are not detected by previous methods (the characteristic classes of Reznikov for example), and improve theorems of Reznikov and Spacil on cohomology-surjectivity to the level of classifying spaces. Joint work with Yasha Savelyev.

#### October 9, 2015: Mohammad Tehrani "Normal crossings divisors and configurations for symplectic topology"

Abstract: In this talk, I will introduce purely symplectic notions of symplectic normal crossing divisor and configuration. These objects generalize the notion of normal crossings in algebraic geometry. Then, I will introduce the notion of “regularization”. Roughly speaking, a regularization is a (compatible set of) symplectic identification(s) of a neighborhood of the divisor in its “normal bundle” with a neighborhood of that in the ambient symplectic manifold; i.e. it is a generalization of "symplectic neighborhood theorem" for smooth symplectic submanifolds. The main result is that every normal crossing divisor/configuration, after some appropriate deformation of the symplectic structure, admits a regularization. As applications, our results give rise to a multifold version of Gompf's symplectic sum construction and a new interpretation of B. Parker's Gromov-Witten invariants relative to normal crossings divisors. (This is a joint work with Mark Mclean and Aleksey Zinger)

#### October 16, 2015: Adam Lowrance "Khovanov homology, chromatic homology, and torsion"

Abstract: Although computations show that Khovanov homology has torsion of many different orders, it is still not understood what causes that torsion to appear. We use a partial isomorphism between the Khovanov homology of a link and the chromatic polynomial categorification of a certain graph associated to that link to study torsion. In particular, we show that Khovanov homology cannot have odd torsion in certain gradings. We also discuss related results and conjectures about torsion in Khovanov and chromatic homology. This is joint work with Radmila Sazdanovic.

#### October 23, 2015: Francesco Lin "Pin(2)-monopole Floer homology"

Abstract: Manolescu has recently introduced new gauge theoretic invariants of rational homology three spheres and used them to disprove the long-standing Triangulation conjecture. In this talk we will discuss how to define these in the alternative setting of Kronheimer-Mrowka's monopole Floer homology (which works for every three manifold), and discuss some of their properties.

#### October 23, 2015: Yoel Groman "Floer theory on open manifolds"

Abstract: I will discuss how to construct Floer theoretic invariants on geometrically bounded open symplectic manifolds without any exactness assumptions (arXiv:1510.04265) . An important class of examples for which this is relevant is the complement of an anticanonical divisor in toric Calabi Yau manifolds. If there will be time I will discuss an application to nearby existence of periodic orbits.

#### October 30, 2015: Sheila (Margherita) Sandon "Floer homology for translated points"

Abstract: A point $q$ in a contact manifold $(M,\xi)$ is said to be a translated point of a contactomorphism $\phi$, with respect to a contact form $\alpha$ for $\xi$, if it is a "fixed point modulo the Reeb flow", i.e. if $q$ and $\phi(q)$ are in the same Reeb orbit and $\phi$ preserves $\alpha$ at $q$. Translated points are key objects to look at when studying contact rigidity phenomena such as contact non-squeezing, orderability of contact manifolds and existence of bi-invariant metrics and quasimorphisms on the contactomorphism group. Based on the notion of translated points, in 2011 I proposed a contact analogue of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms. In my talk I will present a proof of this conjecture under the assumption that there are no closed contractible Reeb orbits, by means of a Floer homology theory for translated points that I am developing ad hoc to study this problem.

#### October 30, 2015: Sheel Ganatra "Functors and relations from Fukaya categories of LG models"

Abstract: The Fukaya category of a Landau-Ginzburg (LG) model W: E --> C, denoted F(E,W), enlarges the Fukaya category of E to include certain non-compact Lagrangians determined by W (for instance, Lefschetz fibrations and their thimbles). I will describe natural functors associated to the Fukaya categories of (E,W) and the general fibre M, and introduce a new Floer homology ring for (E,W). Using these, I will explain two new results: (a) a generation criterion for F(E,W), in the sense of Abouzaid/AFOOO, and (b) exact triangles of functors, one each in F(E,W) and F(M). First applications include stability and generation results for Fukaya categories and a new proof of exact sequences for fibered twists. This is joint work (in preparation) with Mohammed Abouzaid.

#### November 6, 2015: "Modular representations with two-block nilpotent central character"

Abstract: We will study the category of modular representations of the special linear Lie algebra with a fixed two-block nilpotent p-character. Building on work of Cautis and Kamnitzer, we construct a categorification of the affine tangle calculus using these categories; the main technical tool is a geometric localization-type result of Bezrukavnikov, Mirkovic and Rumynin. Using this, we give dimension formulae for the irreducible modules, and a description of the Ext algebra. This Ext algebra is an "annular" analogue of Khovanov's arc algebra, and can be used to give an extension of Khovanov homology to links in the annulus.

#### November 6, 2015: Matt Hogankamp "Categorical diagonalization of the full twist"

Abstract: I will discuss recent joint work with Ben Elias in which we introduce a theory of diagonalization of functors. Our main application is the diagonalization of the the Rouquier complex associated to full-twist braid, acting on the category of Soergel bimodules. The "eigenprojections" yield categorified Young symmetrizers. I will discuss connections to Khovanov homology and a beautiful recent conjecture of Gorsky-Rasmussen which relates these categorical projections to the flag Hilbert scheme.

#### November 12, 2015: Paul Wedrich "Deformations of type A link homologies"

Abstract: I will start by explaining how deformations help to answer two important questions about the family of (colored) sl(N) link homology theories: What geometric information about links do they contain? What relations exist between them? I will recall Lee's deformation of Khovanov homology and sketch how it generalizes to the case of colored sl(N) link homology. The result is a decomposition theorem for deformed colored sl(N) link homologies, which leads to new spectral sequences between various type A link homologies and to new concordance invariants in the spirit of Rasmussen's s-invariant. Part of this is joint work with David E. V. Rose.

#### November 13, 2015: Dmitry Tonkonog "Circle actions and the Fukaya category"

Abstract: The talk will be about Fukaya algebras of monotone Lagrangian submanifolds which have a Hamiltonian circle symmetry. Namely, I will compute a "topological piece" of the closed-open string map into the Hochschild cohomology of such Lagrangians, and talk about applications: split-generation and non-formality results for real Lagrangian submanifolds of some toric varieties.

#### November 20, 2015: Soren Galatius "Tautological rings for high-dimensional manifolds"

Abstract: To each fiber bundle $f: E \to B$ whose fibers are closed oriented manifolds of dimension $d$ and each polynomial $p \in H^*(BSO(d))$ there is an associated "tautological class" $\kappa_p \in H^*(B)$ defined by fiberwise integration. The set of polynomials in these classes which vanish for all bundles whose fibers are oriented diffeomorphic to $M$ forms an ideal $I_M \subset \mathbb{Q}[\kappa_p]$ and the quotient ring $R_M = \mathbb{Q}[\kappa_p]/I_M$ is the "tautological ring" of $M$. In this talk I will discuss some recent results about the structure and particularly Krull dimension of this ring for various $M$. This is joint work with Ilya Grigoriev and Oscar Randal-Williams.

#### December 3, 2015: Krzysztof Putyra "On categorical traces and homology for links in a solid torus"

Abstract: A trace of a category is the set of its endomorphisms considered up to conjugation. For example, the trace of the category of tangles is formed by links in a solid torus: every such a link is a closure of a tangle and closures of two tangles coincide if and only if the tangles agree up to conjugation. The categorical trace is functorial: a functor between categories induces a map between their traces. In particular, we can obtain invariants of links in a solid torus from functors on the category of tangles. In my talk I will discuss the trace of the tangle invariant due to Chen and Khovanov, which recovers the annular sl(2) homology defined by Asaeda, Przytycki, and Sikora. This construction almost immediately equips the APS homology with an action of sl(2). With a small modification of the trace construction one can quantize the homology to obtain a sequence of representations of quantum sl(2). Joint with Anna Beliakova.

#### December 4, 2015: Kyler Siegel "Subflexible symplectic manifolds"

Abstract: After reviewing some recent developments in symplectic flexibility, I will introduce a strange class of symplectic manifolds which are sub-level sets of flexible Stein manifolds but are not themselves flexible. These are examples of exotic symplectic manifolds with trivial symplectic cohomology. I will explain how to detect their nonflexibility using a twisted version of symplectic cohomology, and how this invariant can often be computed using Lefschetz presentations and Fukaya categories with B-fields. This is partly based on joint work with Emmy Murphy.

## 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.