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

Our e-mail list.

 Date Speaker Title Jan. 22 10:30 Math 520 Geordie Williamson (Max-Planck-Institut für Mathematik in Bonn) Lusztig's conjecture and the Hecke category Jan. 29 Nathan Dowlin (Princeton University) Cube of resolutions complexes for Khovanov-Rozansky homology and knot Floer homology Feb. 5 Baptiste Chantraine (Université de Nantes) Positive Legendrian isotopies and Floer theory Feb. 11 IAS Baptiste Chantraine, Lenny Ng, John Etnyre Workshop on Legendrians and Augmentations Feb. 12 IAS Emmy Murphy, Roger Casals, Lisa Traynor Workshop on Legendrians and Augmentations (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Feb. 16 1pm Math 622 Frol Zapolsky (University of Haifa) On the contact mapping class group of the prequantization space over the $A_m$ Milnor fiber Feb. 19 Lenny Ng (Duke University) Toward a contact Fukaya category Feb. 26 10:30 Math 520 Robert Castellano (Columbia University) Gromov-Witten moduli spaces and Kuranishi atlases Feb. 26 Samuel Lisi (University of Mississippi) Some contact 5-manifolds and symplectic fillings Mar. 4 Sheng-Fu Chiu (Northwestern University) From microlocal category to contact non-squeezability Mar. 11 10:30 Math 520 Ivan Ip (Kyoto University) Positive Casimir and Central Characters of Split Real Quantum Groups Mar. 11 David Treumann (Boston College) Perfect schemes from Frobenius-twisted Floer theory Mar. 17-20 Conference in Honor of Dusa McDuff's 70th Birthday Mar. 25 Ken Baker (University of Miami) Cable space surgeries via jointly primitive presentations of knots Apr. 1 10:30 Math 520 Mark McLean (Stony Brook University) Log canonical threshold and Floer homology of the monodromy (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Apr. 1 Renato Vianna (University of Cambridge) Infinitely many monotone Lagrangian tori in del Pezzo surfaces (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Apr. 8 Georgios Dimitroglou Rizell (University of Cambridge) Classification results for two-dimensional Lagrangian tori Apr. 8-10 The 31st Annual Geometry Festival at Princeton University Apr. 15 Artan Sheshmani (MIT/Ohio State University/Kavli IPMU) On proof of S-duality modularity conjecture over compact Calabi-Yau threefolds (Quintic case) Apr. 22 Jonathan Hanselman (University of Texas at Austin) Bordered Floer homology via immersed curves Apr. 29 10:30 Math 520 Xin Jin (Northwestern University) Nadler-Zaslow correspondence without Floer theory Apr. 29 Inanc Baykur (University of Massachusetts Amherst) Small symplectic and exotic 4-manifolds via positive factorizations May 6 Math 507 Brett Parker (Australian National University) Tropical gluing formula for Gromov-Witten invariants

# Abstracts

#### January 22, 2016: Geordie Williamson "Lusztig's conjecture and the Hecke category"

Abstract: Let $G$ be a connected reductive algebraic group (for example $G = GL_n$). We consider the question of determining the irreducible algebraic $G$-modules. If $G$ is defined over a field of characteristic zero the story is just like that for compact groups (parameterisation by dominant weights, Weyl character formula, complete reducibility etc.). If $G$ is defined over a field of positive characteristic some aspects are similar to in characteristic zero (parameterisation by dominant weights) but the character question is very complicated. There is an answer valid in large characteristics given by a formula of Lusztig. I will talk about a conceptual approach to this question via categorification and the Hecke category ("Soergel bimodules").

#### January 29, 2016: Nathan Dowlin "Cube of resolutions complexes for Khovanov-Rozansky homology and knot Floer homology"

Abstract: I will compare the oriented cube of resolutions constructions for Khovanov-Rozansky homology and knot Floer homology. Manolescu conjectured that for singular diagrams (or trivalent graphs) the HOMFLY-PT homology and knot Floer homology are isomorphic - I will show that this conjecture is equivalent to a certain spectral sequence collapsing. This will also lead to a recursion formula for the HOMFLY-PT homology of singular diagrams that categorifies Jaeger's composition product formula.

#### February 11, 2016: Baptiste Chantraine "Positive Legendrian isotopies and Floer theory"

Abstract: In a cooriented contact manifold, a positive Legendrian isotopy is a Legendrian isotopy evolving in the positive transverse direction to the contact plane. Their global behavior differs from the one of Legendrian isotopy and is closer to the one of propagating waves. In this talk I will explain how to use information in the Floer complex associated to a pair of Lagrangian cobordisms (recently constructed in a collaboration with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko) to give obstructions to certain positive loops of some Legendrian submanifolds. This will recover previously known obstructions and exhibit more examples. This is work in progress with V. Colin and G. Dimitroglou Rizell.

#### February 16, 2016: Frol Zapolsky "On the contact mapping class group of the prequantization space over the $A_m$ Milnor fiber"

Abstract: The contact mapping class group of a contact manifold $V$ is the set of contact isotopy classes of its contactomorphisms. When $V$ is the $2n$-dimensional ($n$ at least 2) $A_m$ Milnor fiber times the circle, with a natural contact structure, we show that the full braid group $B_{m+1}$ on $m+1$ strands embeds into the contact mapping class group of $V$. We deduce that when $n=2$, the subgroup $P_{m+1}$ of pure braids is mapped to the part of the contact mapping class group consisting of smoothly trivial classes. This solves the contact isotopy problem for $V$. The construction is based on a natural lifting homomorphism from the symplectic mapping class group of the Milnor fiber to the contact mapping class group of $V$, and on a remarkable embedding of the braid group into the former due to Khovanov and Seidel. To prove that the composed homomorphism remains injective, we use a variant of the Chekanov-Eliashberd dga for Legendrian links in $V$. This is joint work with Sergei Lanzat.

#### February 19, 2016: Lenny Ng "Toward a contact Fukaya category"

Abstract: I will describe some work, currently in a very early stage, regarding a version of the derived Fukaya category for contact 1-jet spaces $J^1(X)$. This category is built from Legendrian submanifolds equipped with augmentations, and the full subcategory corresponding to a fixed Legendrian submanifold $\Lambda$ is the augmentation category $Aug(\Lambda)$, which I will attempt to review. The derived Fukaya category is generated by unknots, with the corollary that all augmentations "come from unknot fillings". I will also describe a potential application to proving that "augmentations = sheaves". This is work in progress with Tobias Ekholm and Vivek Shende, building on joint work with Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow.

#### February 26, 2016: Robert Castellano "Gromov-Witten moduli spaces and Kuranishi atlases"

Abstract: Constructing virtual fundamental classes on moduli spaces of J-holomorphic curves is one of the fundamental challenges in symplectic topology. Kuranishi atlases were introduced by McDuff and Wehrheim to resolve some of the issues in this field. In this talk I will give a brief overview of Kuranishi atlases and show that genus zero Gromov-Witten moduli spaces admit a Kuranishi atlas that is sufficient to define a fundamental class in any virtual dimension. The key step for this is the proof of a stronger gluing theorem.

#### February 26, 2016: Sam Lisi "Some contact 5-manifolds and symplectic fillings"

Abstract: I will discuss some constructions of contact 5-manifolds and give some partial results on symplectic fillability. In particular, we will consider some of Bourgeois’s contact structures on $M \times T^2$, and see that they are very sensitive to the page of the open book decomposition used to construct them, but much less so to the monodromy. This talk is partially about joint work with Van Horn-Morris and Wendl, and partially about joint work with Niederkrüger and Perisic.

#### March 4, 2016: Sheng-Fu Chiu "From microlocal category to contact non-squeezability"

Abstract: In this talk, we will adopt Tamarkin's notion of microlocal category (based on Kashiwara-Schapira's sheaves on manifolds) to the setting of contact topology and define a contact isotopy invariant similar to contact homology when the extra dimension is a circle. We apply this invariant to describe a contact non-squeezing phenomenon proposed by Eliashberg-Kim-Polterovich.

#### March 11, 2016: Ivan Ip "Positive Casimir and Central Characters of Split Real Quantum Groups "

Abstract: The notion of the positive representations was introduced in a joint work with Igor Frenkel as a new research program devoted to the representation theory of split real quantum groups. Explicit construction of the these irreducible representations have been made corresponding to classical Lie type. In this talk, I will discuss the action of the generalized Casimir operators, which is important to understand the tensor product decomposition of these representations. These operators are shown to admit positive eigenvalues, and that their image defines a semi-algebraic region bounded by real points of the discriminant variety.

#### March 11, 2016: David Treumann "Perfect schemes from Frobenius-twisted Floer theory"

Abstract: I would like to find some symplectic constructions that are mirror to nontrivial algebraic geometry in positive characteristic. For example, is there any sense in which an elliptic curve over $\mathbb F_p$ parametrizes Lagrangian circles in a symplectic torus? I don't know. But I will explain how Deligne-Lusztig varieties (for example, the elliptic curve at $j = 0$ in characteristic 2) parametrize Lagrangian surfaces in $S^1 \times \mathbb R^3$. The surfaces are decorated in a way to make them compatible with a mild global structure on the 4-manifold, that maybe should remind you of a $B$-field. There is an interesting limitation of the picture: the functor-of-points of the moduli problem (parametrizing Lag surfaces) can only be evaluated at perfect rings. I hope that prior understanding of the jargon in the abstract will not be necessary to follow the talk.

#### March 25, 2016: Ken Baker "Cable space surgeries via jointly primitive presentations of knots"

Abstract: We describe a novel scheme for creating integral surgeries from 3-manifolds with two torus boundary components to cable spaces. Using this scheme we recover the 22 examples of asymmetric, hyperbolic, 1-cusped, 3-manifolds with two lens space fillings of Dunfield-Hoffman-Licata and many, many more. This scheme also produced the recent family of tunnel number 2 knots in the Poincare homology sphere with lens space surgeries. Time permitting, we'll also discuss related constructions of new surgeries between pairs of reducible manifolds and between lens spaces and reducible manifolds. Joint in part with Hoffman and Licata.

#### April 1, 2016: Mark McLean "Log canonical threshold and Floer homology of the monodromy"

Abstract: The log canonical threshold of a hypersurface singularity is an important invariant which appears in many areas of algebraic geometry. For instance it is used in the minimal model program, has been used to prove vanishing theorems, find Kahler Einstein metrics and it is related to the growth of solutions mod $p^k$. We show how to calculate the log canonical threshold and also the multiplicity of the singularity using Floer homology of iterates of the monodromy map.

#### April 1, 2016: Renato Vianna "Infinitely many monotone Lagrangian tori in del Pezzo surfaces"

Abstract: We will describe how to get almost toric fibrations for all del Pezzo surfaces (endowed with monotone symplectic form), in particular for $\mathbb C P^2\# k\,\overline{\mathbb C P^2}$ for $4 \le k \le 8$, where there is no toric fibrations. From there, we will be able to construct infinitely many monotone Lagrangian tori. We are able to prove that these tori give rise to infinitely many symplectomorphism classes in $\mathbb C P^2 \# k\,\overline{\mathbb C P^2}$ for $0 \le k \le 8$, $k \ne 2$, and in $\mathbb C P^1 \times \mathbb C P^1$. Using the recent work of Pascaleff-Tonkonog one can conclude the same for $\mathbb C P^2 \# 2\,\overline{\mathbb C P^2}$. Some Markov like equations appear. These equations also appear in the work of Haking-Porokhorov regarding degeneration of surfaces to weighted projective spaces and on the work of Karpov-Nogin regarding $3$-block collection of exceptional sheaves in del Pezzo surfaces.

#### April 8, 2016: Georgios Dimitroglou Rizell "Classification results for two-dimensional Lagrangian tori"

Abstract: We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone $S^2 \times S^2$. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.

#### April 15, 2016: Artan Sheshmani "On proof of S-duality modularity conjecture over compact Calabi-Yau threefolds (Quintic case)"

Abstract: I will talk about an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Together with Amin Gholampour we use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture. Our work is based on our earlier results with Richard Thomas and Yukinobu Toda, which I will also discuss as further ingredients, needed for the final proof.

#### April 22, 2016: Jonathan Hanselman "Bordered Floer homology via immersed curves"

Abstract: Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. For a large class of three-manifolds with torus boundary, I will present a geometric interpretation of these invariants in terms of immersed curves on the boundary torus. In this setting, the pairing theorem in bordered Floer homology can be reinterpreted in terms of intersection between curves. As one application, we can use this new interpretation of bordered Floer homology to recover and extend a recent result concerning toroidal L-spaces. This is joint work with Jake Rasmussen and Liam Watson.

#### April 29, 2016: Xin Jin "Nadler-Zaslow correspondence without Floer theory"

Abstract: The Nadler-Zaslow correspondence assigns every exact Lagrangian brane in the cotangent bundle of a manifold $M$ a constructible sheaf on $M$. The construction follows from Floer theory. In this talk, I will present a purely sheaf-theoretic way to realize the correspondence, which has the benefit that it can be generalized to have coefficients in a ring spectrum. Along the way, we will see that the theory of microlocal sheaves gives a natural way of understanding the "brane obstructions", which turns out to be closely related to the J-homomorphism. This is joint work with David Treumann.

#### April 29, 2016: Inanc Baykur "Small symplectic and exotic 4-manifolds via positive factorizations"

Abstract: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes (joint work with Mustafa Korkmaz) which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.

#### May 6, 2016: Brett Parker "Tropical gluing formula for Gromov-Witten invariants"

Abstract:I will describe why tropical curves appear whenever we study holomorphic curves in normal crossing (or log smooth) degenerations, and give a gluing formula for Gromov-Witten invariants which involves a sum of relative Gromov-Witten invariants over tropical curves.

## 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.