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 Title Sept. 6, 1pm Guangbo Xu (Texas A&M) Anti-Self-Dual equation over certain noncompact 4-manifolds and the SO(3) Atiyah--Floer conjecture Sept. 13, 10:30am Umut Varolgunes (Stanford) Applications of relative invariants in the context of symplectic SC divisors with Liouville complement Sept. 13, 1pm Daniel Tubbenhauer (University of Zurich) 2-representations of Soergel bimodules Sept. 20, 10:30am You Qi (Caltech) On a tensor product categorification at prime roots of unity Sept. 20, 1pm Yasuyoshi Yonezawa (Nagoya University) Braid group actions from categorical Howe duality on a category of matrix factorizations and a bimodule category of deformed Webster algebra Sept. 27, 1pm Sara Tukachinsky (IAS) Open WDVV equations Oct. 4, 10:30am Daniel Kaplan (Fields Institute) Formality of Multiplicative Preprojective Algebras Oct. 4, 1pm Vardan Oganesyan (Stony Brook) Monotone Lagrangian submanifolds and toric topology Oct. 11, 10:30am Xiao Zheng (Boston University) Disc potentials of equivariant Lagrangian Floer theory Oct. 11, 1pm Paolo Ghiggini (Nantes) Liouville nonfillability of RP^5 Oct. 18, 10:30am Agustin Moreno (Augsburg) Bourgeois contact structures: tightness, fillability and applications. Oct. 18, 1pm Mariano Echeverria (Rutgers University) A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori Oct. 25, 10:30am Abigail Ward (Harvard University) Homological mirror symmetry for elliptic Hopf surfaces Oct. 25, 1pm Yingdi Qin (Harvard University) Coisotropic branes on symplectic tori Nov. 1, 10:30am Lev Tovstopyat (Boston College) Obstructing Lagrangian link cobordisms via knot Floer homology Nov. 1, 1pm Nitu Kitchloo (Johns Hopkins University) Symmetry Breaking and Link homologies Nov. 8, 10:30am Chris Gerig (Harvard University) Probing 4-manifolds with near-symplectic forms Nov. 8, 1pm Allen David Boozer (UCLA) Computer Bounds for Kronheimer-Mrowka Foam Evaluation Nov. 15, 1pm McKee Krumpak (Brandeis University) Twisted monopole h-invariants Nov. 22, 10:30am Jun Li (University of Michigan) Smooth and symplectic isotopy on rational 4-manifolds Nov. 22, 1pm Dan Cristofaro-Gardiner (UCSC / IAS) Two or infinity Nov. 29 Thanksgiving! Dec. 6, 10:30am Artem Kotelskiy (Indiana University) Immersed curves in Khovanov homology Dec. 6, 1pm Tasos Moulinos (Toulouse) A universal HKR theorem

# Abstracts

#### September 6th, 2019: Guangbo Xu (Texas A&M) " Anti-Self-Dual equation over certain noncompact 4-manifolds and the SO(3) Atiyah--Floer conjecture "

Abstract: Consider the anti-self-dual equation over the product of the real line and a three-manifold with cylindrical end, with gauge group being SO(3). I will explain the proof of a Gromov--Uhlenbeck type compactness result for this equation. This is the first step towards constructing a natural bounding cochain for the symplectic side of the SO(3) Atiyah--Floer conjecture.

#### September 13th, 2019: Umut Varolgunes (Stanford University) " Applications of relative invariants in the context of symplectic SC divisors with Liouville complement "

Abstract: I will start by introducing the elementary notions of SH-visible and SH-full subsets, which are analogous to Entov-Polterovich's heavy and superheavy subsets. Then, I will sketch the proof that in the c_1(M)=0 case, the skeleton of a Liouville domain as appeared in the title is SH-full, and explore some consequences of this (this part is inspired heavily by M. McLean's work). Finally, I will give a speculative discussion about what can happen if c_1(M)=0 is not assumed. This is joint work with D. Tonkonog.

#### September 13th, 2019: Daniel Tubbenhauer (University of Zurich) " 2-representations of Soergel bimodules "

Abstract: The representation theory of Hecke algebra is unambiguous in mathematics and beyond. In this talk I will give a survey about a categorification of this theory, which we call 2-representation of Soergel bimodules. (Joint with Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang.)

#### September 20th, 2019: You Qi (Caltech) " On a tensor product categorification at prime roots of unity "

Abstract: Motivated by finding a categorical analogue of conformal blocks, we explain a formalism of extending a given categorical quantum group representation on a Weyl module to a certain tensor product representations. In particular, equipped with p-differential graded structures, the machinery gives rise to a categorification of certain tensor product representations of Weyl modules at prime roots of unity. This is based on joint work and work in progress with M. Khovanov and J. Sussan.

#### September 20th, 2019: Yasuyoshi Yonezawa (Nagoya University) " Braid group actions from categorical Howe duality on a category of matrix factorizations and a bimodule category of deformed Webster algebra "

Abstract: Many link homology theories can be understood as categorical braid group actions arising from a categorification of Howe duality. In the theory of categorified quantum groups in type A, we can define a complex that categorifies the Lusztig's braid group action on representations of the quantum group of type A. Using categorical Howe duality, the complex induces several categorifications of R-matrix in representations of the quantum group of type A, for instance, Khovanov-Rozansky sl(n) homology, the complex of Soergel bimodules. Here I talk about the categorical skew Howe duality via the category of matrix factorizations (joint work with Mackaay) and the categorical symmetric Howe duality via the bimodule category of deformed Webster algebras (joint work with Khovanov, Lauda and Sussan).

#### September 27th, 2019: Sara Tukachinsky (IAS) " Open WDVV equations "

Abstract: In a work from 2016, joint with Jake Solomon, we use Fukaya A_\infty algebras and bounding chains to define genus zero open Gromov-Witten invariants. These invariants count configurations of pseudoholomorphic disks in a symplectic manifold X with boundary conditions in a Lagrangian submanifold L. However, the construction is rather abstract. Nonetheless, in recent work with Jake Solomon, we find that the superpotential that generates these invariants has many properties that enable calculations. For instance, we establish a vanishing result that limits the amount of boundary constraints if L is homologically nontrivial. Other properties include a wall-crossing formula and open Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In fact, for (X,L)=(CP^n,RP^n), all possible values of the open Gromov-Witten invariants are computable by a recursive process. The open WDVV equations can be further reinterpreted as the associativity of a quantum product on relative cohomology H^*(X,L).

#### October 4th, 2019: Daniel Kaplan (Fields Institute) " Formality of Multiplicative Preprojective Algebras "

Abstract: I'll begin by motivating the study of derived multiplicative preprojective algebras from the perspective of Fukaya categories of plumbed cotangent bundles of spheres, following Etgü and Lekili. Next I'll turn to the question of formality: can one recover the Fukaya category from the category of modules over the homology of such algebras, as opposed to the clunkier dg-category of dg-modules for the entire dg-algebra? The answer turns out to be no in the ADE Dynkin case, yes for quivers containing a cycle, and maybe (but probably yes with analogy to the additive setting) in all other cases. The method of proof of formality in the case of quivers containing a cycle (joint work with Travis Schedler) may be of independent interest.

#### October 4th, 2019: Vardan Oganesyan (Stony Brook) " Monotone Lagrangian submanifolds and toric topology "

Abstract: Let N be the total space of a bundle over some k-dimensional torus with fibre Z, where Z is a connected sum of sphere products. It turns out that N can be embedded into C^n and CP^n as a monotone Lagrangian submanifold. It is possible to construct embeddings of N with different minimal Maslov numbers and get submanifolds which are not Lagrangian isotopic. Also, we will discuss restrictions on Maslov class of monotone Lagrangian submanifolds of C^n. We will show that in certain cases our examples realize all possible minimal Maslov numbers. In addition, we can show that some of our embeddings are smoothly isotopic but not isotopic through Lagrangians. (joint with Yuhan Sun)

#### October 11th, 2019: Xiao Zheng (Boston University) " Disc potentials of equivariant Lagrangian Floer theory "

Abstract: In this talk, I will introduce an equivariant mirror construction using a Morse model of equivariant Lagrangian Floer theory, formulated in a joint work with Kim and Lau. In case of semi-Fano toric manifold, our construction recovers the $T$-equivariant Landau-Ginzburg mirror found by Givental. For toric Calabi-Yau manifold, the equivariant disc potentials of certain immersed Lagrangians are closely related to the open Gromov-Witten invaraints of Aganagic-Vafa branes, which were studied by Katz-Liu, Graber-Zaslow, Fang-Liu-Zong and many others using localization techniques. The later result is a work in progress joint with Hong, Kim and Lau.

#### October 11th, 2019: Paolo Ghiggini (Nantes) " Liouville nonfillability of RP^5 "

Abstract: I will prove that RP^5 with its standard contact structure is not the boundary of a Liouville manifold. The proof is inspired by McDuff's classification of fillings of RP^3. This is a joint work with Klaus Niederkruger.

#### October 18th, 2019: Agustin Moreno (Augsburg) " Bourgeois contact structures: tightness, fillability and applications. "

Abstract: Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.

#### October 18th, 2019: Mariano Echeverria (Rutgers University) " A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori "

Abstract: Given a knot K inside an integer homology sphere Y , the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen.

Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically non trivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori.

This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. In particular, when (X, T ) is obtained from a self-concordance of a knot (Y,K) satisfying an admissibility condition, these Donaldson invariants are related to the Lefschetz number of an Instanton Floer homology for knots which we construct. Moreover, from these Floer groups we obtain Frøyshov invariants for knots which allows us to assign a Frøyshov invariant to an embedded torus whenever it arises from such a self-concordance.

#### October 25th, 2019: Abigail Ward (Harvard University) " Homological mirror symmetry for elliptic Hopf surfaces "

Abstract: We show that homological mirror symmetry is a phenomenon that exists beyond the world of Kähler manifolds by exhibiting HMS-type results for a family of complex surfaces which includes the classical Hopf surface (S^1 x S^3). Each surface S we consider can be obtained by performing logarithmic transformations to the product of P^1 with an elliptic curve. We use this fact to associate to each S a mirror "non-algebraic Landau-Ginzburg model" and an associated Fukaya category, and then relate this Fukaya category and the derived category of coherent analytic sheaves on S.

#### October 25th, 2019: Yingdi Qin (Harvard University) " Coisotropic branes on symplectic tori "

Abstract: Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. Kapstin and Orlov conjecture that coisotropic branes should be included into the Fukaya category from a physics view point. In this talk, I will construct for linear symplectic tori the version of Fukaya category including coisotropic branes and show that the usual Fukaya category embeds fully faithfully into it.

#### November 1st, 2019: Lev Tovstopyat (Boston College) " Obstructing Lagrangian link cobordisms via knot Floer homology "

Abstract: I'll explain how an invariant of Legendrian links in knot Floer homology can be used to obstruct the existence of decomposable Lagrangian link cobordisms in a very general setting. The argument involves braiding the ends of the cobordism about open books and appealing to an algebraic property of the Legendrian invariant called co-multiplication. Much of the talk will be spent describing the contact geometric ingredients that go into the argument.

#### November 1st, 2019: Nitu Kitchloo (Johns Hopkins University) " Symmetry Breaking and Link homologies "

Abstract: Given a compact connected Lie group G endowed with root datum, and an element w in the corresponding Artin braid group for G, we describe a filtered G-equivariant stable homotopy type called strict Broken Symmetries, sBSy(w). As the name suggests, sBSy(w) is constructed from pincipal G-connections on a circle, whose holonomy is broken between consecutive sectors in a manner prescribed by a presentation of w. Specializing to the case of the unitary group G=U(r), we show that sBSy(w) is an invariant of the link L obtained by closing the r-stranded braid w. As such, we denote it by sBSy(L). We may therefore obtain (group valued) link homology theories as terms in a spectral sequence obtained on applying suitable U(r)-equivariant cohomology theories E to sBSy(L). We offer two examples of such theories. In the first example, we take E to be Borel-equivariant singular cohomology. In this case, one recovers an unreduced,integral form of the Triply-graded link homology as the E_2-term. In the next example, we apply a version of an equivariant K-theory known as Dominant K-theory, which is built from level n representations of the loop group of U(r). In this case, the E_2-term recovers a deformation of sl(n)-link homology, and has the property that its value on the unknot is the Grothendieck group of level n-representations of the loop group of U(1).

#### November 8th, 2019: Chris Gerig (Harvard University) " Probing 4-manifolds with near-symplectic forms "

Abstract: Most closed 4-manifolds don't admit symplectic forms, but most admit "near-symplectic" forms that are symplectic away from some embedded circles. This provides a gateway from the symplectic world to the non-symplectic world, and just like the Seiberg-Witten invariants there are counts of J-holomorphic curves that are compatible with the near-symplectic form. Although (potentially exotic) 4-spheres don't even admit near-symplectic forms, there is still a way to bring in near-symplectic techniques, and I will describe my attempt(s) to probe them using J-holomorphic curves.

#### November 8th, 2019: Allen David Boozer (UCLA) " Computer Bounds for Kronheimer-Mrowka Foam Evaluation "

Abstract: Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. One outgrowth of their approach is the definition of a functor J^flat from the category of webs and foams to the category of integer-graded vector spaces over the field of two elements. Of particular interest is the relationship between the dimension of J^flat(K) for a web K and the number of Tait colorings Tait(K) of K. I describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of J^flat(K) for a given web K, in many cases determining these quantities uniquely.

#### November 15th, 2019: McKee Krumpak (Brandeis University) " Twisted monopole h-invariants "

Abstract: I will describe a generalization of Froyshov’s h-invariant, defined for a pair (Y, A) where Y is an arbitrary closed 3-manifold and A is a subgroup H^1(Y, Z). This involves a calculation of the bar flavor of monopole Floer homology with coefficients in a local system associated to A. Time permitting, I will discuss a conjectural relationship with Mrowka-Ruberman-Saveliev’s Casson-type invariant of 4-manifolds with the homology of S^1×S^3

#### November 22nd, 2019: Jun Li (University of Michigan) " Smooth and symplectic isotopy on rational 4-manifolds "

Abstract: We study rational 4-manifolds and their symplectomorphism groups. Analogous to the 2-dimensional case, one can think about the symplectic Torelli group and symplectic mapping class group, where the former is the subgroup of the symplectomorphism group fixing homology and the latter is the symplectomorphism group mod symplectic isotopy. We’ll show how the symplectic mapping class groups of rational 4-manifolds are related to braid groups. Also, McDuff-Salamon asked whether any Torelli symplectomorphism on a rational 4-manifold is smoothly isotopic to identity. And we answer it in the positive. This is based on joint works with T-J Li and Weiwei Wu.

#### November 22nd, 2019: Dan Cristofaro-Gardiner (UCSC / IAS) " Two or infinity "

Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by C. Taubes in 2007. I will discuss joint work showing that for a wide class of Reeb vector fields on closed three-manifolds, there are either two, or infinitely many, distinct closed orbits. I will also say a few words about how one might extend this result to all Reeb vector fields on closed three-manifolds.

#### December 6th, 2019: Artem Kotelskiy (Indiana University) " Immersed curves in Khovanov homology "

Abstract: Consider a 2-sphere S intersecting a knot K in 4 points. This defines decomposition of a knot into two 4-ended tangles. We will show that Khovanov homology Kh(K), and its deformation due to Bar-Natan, are isomorphic to wrapped Lagrangian Floer homology of a pair of specifically constructed immersed curves on the dividing 4-punctured sphere S. This result is analogous to immersed curves description of bordered Heegaard Floer homology and knot Floer homology. The key step will be constructing a tangle invariant in the form of a chain complex over a certain algebra B (deformation of Khovanov's arc algebra), and showing that algebra B embeds into the wrapped Fukaya category of the 4-punctured sphere. As an application, we will prove that Conway mutation preserves Rasmussen's s-invariant of knots. This is joint work with Liam Watson and Claudius Zibrowius.

#### December 6th, 2019: Tasos Moulinos (Toulouse) " A universal HKR theorem "

Abstract: The Hochschild-Konstant-Rosenberg theorem is a classic result identifying the Hochschild homology of a commutative ring with differential forms. In characteristic zero, this can be promoted to an equivalence at the level of differential graded algebras, giving rise to the Hodge decomposition on Hochschild homology. Moreover, via this description, one interprets the de Rham differential on differential forms as the natural $S^1$-action on Hochschild homology. In fact, these constructions globalize, and so one obtains an equivalence of the derived loop space $\mathcal{L}X$ (whose algebra of functions is Hochschild homology) with the shifted tangent bundle $T[−1]X = Spec(Sym(\Omega_X[1]))$.

In this talk, I will explain how this fails in nonzero characteristic and will describe the discrepancy in the form of a natural filtration on the Hochschild homology complex, whose associated graded is derived de Rham cohomology. In characteristic zero, this filtration naturally splits. I will review the relevant notions from derived algebraic geometry used to construct this filtration before explaining the construction in greater depth. Time permitting, I will discuss several applications of this construction and its variants, to a notion of shifted symplectic structures in characteristic p, and to a potentially very clean conceptual reinterpretation of the Witten genus in the algebro-geometric setting. Much of the above is joint work with Marco Robalo and Bertrand Töen.

## 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.