The SGGTC seminar meets on Fridays in Math 520 from 10:30-11:30am and in Math 407 from 1-2pm, unless noted otherwise (in red).

Our e-mail list.

 Date Speaker(s) Title Sep. 7, 10:30am Bohan Fang (Peking University) Holomorphic anomaly equation and modularity from topological recursion Sep. 7, 1pm Melissa Zhang (Boston College) Localization in Khovanov homology Sep. 14, 10:30am Sam Nariman (Northwestern) On the moduli space of flat symplectic surface bundles Sep. 14, 1pm Matt Stoffregen (MIT) An infinite-rank summand of the homology cobordism group Sep. 21, 10:30am Jeff Hicks (Berkeley) Tropical Geometry and Mirror Symmetry Sep. 21, 1pm Nathan Dowlin (Dartmouth) Link invariants related to knot Floer homology and Khovanov homology Sep. 28, 10:30am Yu-Wei Fan (Harvard) Systoles, Special Lagrangians, and Bridgeland stability conditions Oct. 12, 1pm Maggie Miller (Princeton) The Price twist and trisections Oct. 19, 10:30am Yusuf Barış Kartal (MIT) Distinguishing fillings using dynamics of Fukaya categories Oct. 19, 1pm John Baldwin (Boston College) Khovanov homology, instantons, and link detection Oct. 26, 10:30am Harold Williams (Davis) Title: Kasteleyn operators from mirror symmetry Oct. 26, 1pm Siqi He (Simons Center) The Kapustin-Witten Equations, Opers and Khovanov Homology Nov. 2, 10:30am Tye Lidman (NCSU) Spineless four-manifolds Nov. 2, 1pm Kevin Sackel (MIT) Convex contact handle decompositions Nov. 9, 10:30am Jianfeng Lin (MIT) The geography problem on 4-manifolds: 10/8 + 4 Nov. 9, 1pm Jamie Conway (Berkeley) Classifying contact structures on hyperbolic 3–manifolds Nov. 23 Thanksgiving! Nov. 30, 10:30am Ben Gammage (Berkeley) Mirror symmetry and representation theory Nov. 30, 1pm Min Hoon Kim (KIAS) The bipolar filtration of topologically slice knots Dec. 7, 10:30am Mike Miller (UCLA) Equivariant instanton homology and group cohomology Dec. 7, 1pm Egor Shelukhin (University of Montreal) Upper bounds on the Lagrangian spectral norm

# Abstracts

#### 10:30am, Sep. 7, 2018: Bohan Fang (Peking University) " Holomorphic anomaly equation and modularity from topological recursion "

Abstract: The Gromov-Witten invariants are expected to satisfy certain set of differential equations called holomorphic anomaly equations (HAE). Eynard-Orantin's topological recursion describes all genus GW invariants for toric CY 3-folds from their mirror curves. I will talk about the case when the mirror curve is genus 1 and and explicitly written in an hyperelliptic form. Then such recursion gives an explicit modular structure of the GW invariants generating function and implies HAE and its functional version (Yamaguchi-Yau). This talk is based on the joint works with Chiu-Chu Melissa Liu and Zhengyu Zong, and with Yongbin Ruan, Yingchun Zhang and Jie Zhou.

#### 1pm, Sep. 7, 2018: Melissa Zhang (Boston College) " Localization in Khovanov homology "

Abstract: We show that if a cyclic group G acts on a link L, then the Lipshitz-Sarkar Khovanov homotopy type of L admits a G-action, and the fixed-point set of this action is the annular Khovanov homotopy type of the quotient link L/G. By applying the classical Smith inequality, we obtain many new rank inequalities. The talk will include an primer on our main technical tool: the Burnside functor construction of Khovanov spectra introduced by Lawson, Lipshitz, and Sarkar. This is joint work with Matthew Stoffregen.

#### 10:30am, Sep. 14, 2018: Sam Nariman (Northwestern) " On the moduli space of flat symplectic surface bundles "

Abstract: There are at least three different approaches to construct characteristic invariants of a flat symplectic bundle. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. For surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes. In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic surface bundles. As an application, we give a homotopy theoretic description of Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.

#### 1pm, Sep. 14, 2018: Matt Stoffregen (MIT) " An infinite-rank summand of the homology cobordism group "

Abstract: This talk explains a generalization of the techniques that Hom introduced to construct an infinite-rank summand of the topologically slice knot concordance group. We generalize Hom's epsilon-invariant to the involutive Heegaard Floer homology constructed by Hendricks-Manolescu. As an application, we see that there is an infinite-rank summand of the homology cobordism group generated by Seifert spaces. The talk will contain a review of involutive Floer homology. This is joint work with Irving Dai, Jen Hom, and Linh Truong.

#### 10:30am, Sep. 21, 2018: Jeff Hicks (Berkeley) " Tropical Geometry and Mirror Symmetry "

Abstract: Homological mirror symmetry predicts that the Fukaya category of a symplectic manifold X can be matched with the derived category of coherent sheaves on a mirror space Y. TheStrominger-Yau-Zaslow conjecture states that X and Y should have dual Lagrangian torus fibrations, and that mirror symmetry can be recovered by reducing the symplectic and complex geometry of X and Y to tropical geometry on the base of the fibration. In this framework, we expect that Lagrangian fibers of X are mirror to skyscraper sheaves of points on Y, and that Lagrangian sections of the fibration are mirror to line bundles on Y. I will explain how to extend these correspondences to tropical Lagrangians in X and sheaves supported on cycles of intermediate dimension on toric varieties.

#### 1pm, Sep. 21, 2018: Nathan Downlin (Dartmouth) " Link invariants related to knot Floer homology and Khovanov homology "

Abstract: Despite the differences in their constructions, knot Floer homology and Khovanov-Rozansky homology seem to have a great deal in common. I will introduce a family of invariants HFK_n on the knot Floer side which are the knot Floer analogs of sl_n homology. These invariants don't readily allow a cube of resolutions construction, but in the n=2 case I will give an algebraically constructed complex which is expected to be quasi-isomorphic to HFK_2. This complex does decompose as an oriented cube of resolutions, and we will show that the E_2 page of the associated spectral sequence is isomorphic to Khovanov homology. Since reduced HFK_2 is isomorphic to delta-graded HFK, this gives a possible construction of the spectral sequence from Khovanov homology to knot Floer homology. Joint with Akram Alishahi.

#### 10:30am, Sep. 28, 2018: Yu-Wei Fan (Harvard) " Systoles, Special Lagrangians, and Bridgeland stability conditions "

Abstract: Loewner's torus systolic inequality states that the least length of a non-contractible loop on a torus can not be too large compare to its volume. We attempt to generalize this inequality from the viewpoint of Calabi-Yau geometry. This naturally leads to the notions of systoles and systolic ratios of Bridgeland stability conditions. We will first recall some backgrounds on mirror symmetry and Bridgeland stability conditions. Then we will study an example of K3 surface, which turns out to be a lattice-theoretic problem.

#### 1pm, Oct. 12, 14, 2018: Maggie Miller " The Price twist and trisections "

Abstract: Let $S$ be an $RP^2$ embedded in a smooth $4$-manifold $X^4$. With some mild conditions, the Price twist is a surgery operation on $S$ that yields a $4$-manifold homeomorphic (but not necessarily diffeomorphic) to $X^4$. In particular, for every $RP^2$ embedded in $S^4$, this operation yields a homotopy $4$-sphere. In this talk, we will understand the Price twist via the theory of trisections. In particular, I will show how to produce an explicit trisection diagram of a Price-surgered $4$-manifold. Much of the talk will be spent reviewing the theory of trisections, including bridge trisections of surfaces in 4-manifolds and relative trisections of 4-manifolds with boundary. This is joint work with Seungwon Kim.

#### 10:30am, Oct. 19, 14, 2018: Yusuf Barış Kartal (MIT) " Distinguishing fillings using dynamics of Fukaya categories "

Abstract: Given a Weinstein domain $M$ and an exact, compactly supported symplectomorphism $\phi$, one can construct a Weinstein domain $T_\phi$- the open symplectic mapping torus. Its boundary is independent of $\phi$; thus, it gives a filling of $\partial(T_0\times M)$ where $T_0$ is the punctured two torus. In this talk, we will outline a method to distinguish the fillings $T_\phi$ and $T_\0\times M$ using the dynamics and deformation theory of their (wrapped) Fukaya categories. This will involve the construction of a mirror symmetry inspired algebro-geometric model related to Tate curve for the Fukaya category of $T_\phi$. We will exploit dynamics on these models to distinguish them.

#### 1pm, Oct. 19, 14, 2018: John Baldwin (Boston College) " Khovanov homology, instantons, and link detection "

Abstract: In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, answering a categorical version of the famous open question of whether the Jones polynomial detects the unknot. Their proof makes use of a spectral sequence relating Khovanov homology with a version of instanton Floer homology for links. Last year, Steven Sivek and I used their spectral sequence together with ideas in sutured manifold theory and contact geometry to prove that Khovanov homology also detects the right- and left-handed trefoils. I'll discuss this result and some of the key elements of its proof. I'll end with some open questions related to link detection (does Khovanov homology detect the Hopf link?) and knot surgery (are knots with SU(2)-abelian surgeries fibered with 3-genus equal to smooth 4-genus?) which we hope to answer in the near future.

#### 10:30am, Oct. 26, 2018: Harold Williams (Davis) " Kasteleyn operators from mirror symmetry "

Abstract: Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathcal{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathcal{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow.

#### 1pm, Oct. 26, 2018: Siqi He (Simons Center) " The Kapustin-Witten Equations, Opers and Khovanov Homology "

Abstract: We will discuss a Witten's gauge theory program to define Jones polynomial and Khovanov homology for knots inside of general 3-manifolds by counting singular solutions to the Kapustin-Witten or Haydys-Witten equations. We will prove that the dimension reduction of the solutions moduli space to the Kapustin-Witten equations can be identified with Beilinson-Drinfeld Opers moduli space. We will also discuss the relationship between the Opers and a symplectic geometry approach to define the Khovanov homology for 3-manifolds. This is joint work with Rafe Mazzeo.

#### 10:30am, Nov. 2, 2018: Tye Lidman (NCSU) " Spineless four-manifolds "

Abstract: Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of four-manifolds homotopy equivalent to surfaces.

#### 1pm, Nov. 2, 2018: Kevin Sackel (MIT) " Convex contact handle decompositions "

Abstract: Convex hypersurfaces have played an important role in the development of three-dimensional contact geometry. Their role in higher dimensions, however, remains mysterious. One method for understanding them is to study convex contact manifolds, which can be thought of as an analogue of a Weinstein structure for a contact manifold, in which convex hypersurfaces appear as level sets of a Morse function. As in the Weinstein setting, convex contact manifolds naturally decompose into handle decompositions. In this talk, we will explore these handle decompositions and describe their relationship with Weinstein open books.

#### 10:30am, Nov. 9, 2018: Jianfeng Lin (MIT) " The geography problem on 4-manifolds: 10/8 + 4 "

Abstract: A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins and XiaoLin Danny Shi and Zhouli Xu.

#### 1pm, Nov. 9, 2018: Jamie Conway (Berkeley) " Classifying contact structures on hyperbolic 3–manifolds "

Abstract: Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, notably not including hyperbolic manifolds. In this talk, I will present a new classification result for contact structures on an infinite family of hyperbolic 3–manifolds arising from Dehn surgery on the figure-eight knot, and how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

#### 10:30am, Nov. 30, 2018: Ben Gammage (Berkeley) " Mirror symmetry and representation theory "

Abstract: We discuss a paradigm which should govern some mirror symmetry equivalences among hyperkähler spaces of interest in representation theory. Thanks to recent work of Ganatra-Pardon-Shende on Fukaya categories of Weinstein manifolds, it is now possible to make explicit computations of Fukaya categories for these spaces. We explain how these tools can be used to prove a mirror symmetry statement conjectured by Bezrukavnikov for the Springer resolution.

#### 1pm, Nov. 30, 2018: Min Hoon Kim (KIAS) " The bipolar filtration of topologically slice knots "

Abstract: The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. In this talk, I prove that every graded quotient of the bipolar filtration has infinite rank. The proof uses higher order amenable Cheeger-Gromov rho invariants and Heegaard Floer d-invariants of infinitely many cyclic branched covers simultaneously. This is joint work with Jae Choon Cha.

#### 10:30am, Dec. 7, 2018: Mike Miller (UCLA) " Equivariant instanton homology and group cohomology "

Abstract: Equivariant instanton homology and group cohomology Abstract: Floer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds must be somehow equivariant - respecting a certain SO(3)-action. We explain how ideas from group cohomology and equivariant algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.

#### 1pm, Dec. 7, 2018: Egor Shelukhin (University of Montreal) " Upper bounds on the Lagrangian spectral norm "

Abstract: We discuss recent developments in establishing uniform bounds on the spectral norm and related invariants in the absolute and relative settings. In particular, we describe new progress on a conjecture of Viterbo asserting such bounds for exact deformations of the zero section in unit cotangent disk bundles. This talk is partially based on joint work with Asaf Kislev.

## 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.