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) TBA Oct. 19, 1pm John Baldwin (Boston College) Khovanov homology, instantons, and link detection Oct. 26, 10:30am Harold Williams (Davis) TBA Oct. 26, 1pm Siqi He (Simons Center) TBA Nov. 2, 10:30am Tye Lidman (NCSU) TBA Nov. 2, 1pm Kevin Sackel (MIT) TBA Nov. 9, 10:30am Jianfeng Lin (MIT) TBA Nov. 9, 1pm Jamie Conway (Berkeley) TBA Nov. 23 Thanksgiving! TBA Nov. 30, 10:30am Ben Gammage (Berkeley) TBA Nov. 30, 1pm Min Hoon Kim (KIAS) The bipolar filtration of topologically slice knots Dec. 7, 10:30am Mike Miller (UCLA) TBA Dec. 7, 1pm Egor Shelukhin (University of Montreal) TBA Dec. 14, 10:30am TBA TBA Dec. 14, 1pm TBA TBA

# 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.

#### 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.

#### 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.

## 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.