The SGGTC seminar meets on Fridays in Math 407, at 10:45 am unless noted otherwise (in red).

Our e-mail list.

 Date Speaker Title Jan. 23 10:45 Caitlin Leverson (Duke University) Legendrian Knots, Augmentations, and Rulings 1:15 Organizational Meeting Math 520 Organizational MeetingJoint with Geometric Topology Seminar Jan. 30 Vladimir Retakh (Rutgers University) Noncommutative triangulations Feb. 6 Pedro Acosta (University of Michigan) Extending the Landau-Ginzburg/Calabi-Yau correspondence to non-Calabi-Yau hypersurfaces in weighted projective space Feb. 13 Andrew Blumberg (UT Austin) K-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace Feb. 20 John Pardon (Stanford University) Existence of Lefschetz fibrations on Stein/Weinstein domains Feb. 27 Francois Charest (Columbia) Stabilizing divisors and Morse-Bott Floer cohomology Mar. 6 Luis Haug (CRM Montreal) Lagrangian cobordisms and Grothendieck groups Mar. 13 10:45 Denis Auroux (UC Berkeley) Two new constructions of monotone Lagrangian tori 2:15 Yanki Lekili (UIC) Math 507 Koszul duality patterns in Floer theory Mar. 20 1:30 Mohammed Abouzaid (Columbia) Princeton, Jadwin A10 Nearby Lagrangians are simply homotopic (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) 2.45 Susan Tolman (UIUC) Princeton, Jadwin A10 Non-Hamiltonian actions with isolated fixed points (Joint Columbia-Princeton-IAS Symplectic Topology Seminar) Mar. 23 10:10 Ciprian Manolescu (UCLA) Math 507 Seiberg-Witten Floer K-theory and its applications Mar. 25 10:45 Barney Bramham (Bochum) Math 528 Floer-like complexes for surfaces, maximally unlinked braids, and finite energy foliations Mar. 27 Eugene Gorsky (Columbia) Torus knots and Cherednik algebras Apr. 3 Roger Casals (ICMAT, Madrid) Overtwisted contact structures Apr. 10 9:30 Sobhan Seyfaddini (MIT) Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants. 10:45 Jake Rasmussen (University of Cambridge) L-Space filling slopes of homology solid tori Apr. 17 Chris Woodward (Rutgers University) Floer theory and the minimal model program Apr. 24 Nate Bottman (MIT) Quilts with figure eight singularity: progress report May 1 Allison Moore (Rice University) Cosmetic crossing changes in thin knots May 1-3 Courant Institute for Mathematical Sciences NYU geometry festival

# Abstracts

#### January 23, 2015: Caitlin Leverson " Legendrian Knots, Augmentations, and Rulings"

Abstract: A Legendrian knot in $\mathbb{R}^3$ with the standard contact structure is a knot for which $dz-ydx=0$. Given a Legendrian knot, one can associate the Chekanov- Eliashberg differential graded algebra (DGA) over $\mathbb{Z}/2$. Fuchs and Sabloff showed there is a correspondence between augmentations to $\mathbb{Z}/2$ of the DGA and rulings of the knot diagram. Etnyre, Ng, and Sabloff showed that one can define a lift of the Chekanov-Eliashberg DGA over $\mathbb{Z}/2$ to a DGA over $\mathbb{Z}[t,t^{-1}]$. This talk will give an extension of the relationship between rulings and augmentations to $\mathbb{Z}/2$ for the DGA over $\mathbb{Z}/2$, to a relationship between rulings and augmentations to a field of the DGA over $\mathbb{Z}[t,t^{-1}]$. No knowledge of the Chekanov-Eliashberg DGA will be assumed.

#### January 30, 2015: Vladimir Retakh, "Noncommutative triangulations "

Abstract: The celebrated Ptolemy relation plays an important role in various studies of triangulated surfaces including hyperbolic geometry, geometrical applications of cluster algebras and so on. We will discuss a noncommutative version of the relation which can be seen as a "categorification" of the classical one. This leads to new noncommutative invariants of the surfaces and provides several examples of the noncommutative Laurent phenomenon. (Joint work with Arkady Berenstein from University of Oregon)

#### February 6, 2015: Pedro Acosta, " Extending the Landau-Ginzburg/Calabi-Yau correspondence to non-Calabi-Yau hypersurfaces in weighted projective space"

Abstract: In the early days of mirror symmetry, physicists noticed a remarkable relation between the Calabi-Yau geometry of a hypersurface in projective space defined by a homogenous polynomial W and the singularity theory of the Landau-Ginzburg model with superpotential W. This relation came to be known as the Landau-Ginzburg/Calabi-Yau correspondence. In this talk, I will explain how this correspondence can be extended to non-Calabi-Yau hypersurfaces in weighted projective space using the recently introduced Fan-Jarvis-Ruan-Witten theory as the mathematical formalism behind Landau-Ginzburg models.

#### February 13, 2015: Andrew Blumberg "K-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace"

Abstract: Our understanding of the algebraic K-theory of the sphere spectrum boils down to studying the fiber of the cyclotomic trace (an analogue of the Chern character), a map from K(S) to a topological analogue of cyclic homology. It turns out that this fiber can in turn be studied in terms of the p-completion map in etale cohomology. The purpose of this talk is to explain this story and describe joint work with Mike Mandell that describes the fiber in terms of a kind of Poincare duality.

#### February 20, 2015: John Pardon, " Existence of Lefschetz fibrations on Stein/Weinstein domains"

Abstract: I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

#### February 27, 2015: Francois Charest, "Stabilizing divisors and Morse-Bott Floer cohomology "

Abstract: We use the technique of stabilizing divisors introduced by Cieliebak-Mohnke to construct (strictly unital) $A_\infty$ algebras of Lagrangians in rational compact symplectic manifolds. The homotopy type of the algebra and the moduli space of solutions to the weak Maurer-Cartan equation is shown to be independent of the choice of perturbation data. As an application, we show that the Floer cohomology of compact Lagrangians that are free transitive orbits of a symplectic group action are weakly unobstructed; this had been shown earlier by Fukaya-Oh-Ohta-Ono in a different foundational system. Joint work with Chris Woodward.

#### March 6, 2015: Luis Haug, "Lagrangian cobordisms and Grothendieck groups"

Abstract: Lagrangian cobordisms (in the sense of recent work of Biran-Cornea) lead to iterated cone decompositions in the derived Fukaya category DFuk(M) of a symplectic manifold, which induces a canonical homomorphism from a naturally defined Lagrangian cobordism group to the Grothendieck group of DFuk(M). I will explain this circle of ideas, and the proof of why this is an isomorphism in the simplest non-trivial case of a torus.

#### March 13, 2015, 10:45 a. m. : Denis Auroux, "Two new constructions of monotone Lagrangian tori "

Abstract: We will discuss some recent constructions of "exotic" Lagrangian tori in simple symplectic manifolds such as $\mathbb{CP}^2$ (work of Renato Vianna) and $\mathbb{R}^6$ that are not Hamiltonian isotopic to previously known examples, inspired by wall-crossing phenomena and mirror symmetry.

#### March 13, 2015, 2:15 p. m. : Yanki Lekili, "Koszul duality patterns in Floer theory."

Abstract: We study symplectic invariants of the open symplectic manifolds X obtained by plumbing cotangent bundles of spheres according to a plumbing tree. We prove that certain models for the Fukaya category F(X) of closed exact Lagrangians in X and the wrapped Fukaya category W(X) are related by Koszul duality. As an application, we give explicit computations of symplectic cohomology for the vast majority of these symplectic manifolds (including the case of $A_n$-Milnor fibres). This is joint work with Tolga Etgü.

#### March 20, 2015, 1:30 p. m.: Mohammed Abouzaid, "Nearby Lagrangians are simply homotopic"

Abstract: This is a report on joint work in progress with T. Kragh, wherein we prove that a closed exact Lagrangian in a cotangent bundle is simply homotopy equivalent to the base. I will explain the two main ingredients of the proof: (i) realising the Whitehead torsion of the projection to the base as the torsion of a Floer theoretic map and (ii) using a large Hamiltonian deformation to deform the Floer complexes in such a way that this torsion can be shown to be trivial by an action filtration argument.

#### March 20, 2015, 2:45 p. m.: Susan Tolman, "Non-Hamiltonian actions with isolated fixed points"

Abstract: Let a circle act symplectically on a closed symplectic manifold $M$. If the action is Hamiltonian, we can pass to the reduced space; moreover, the fixed set largely determines the cohomology and Chern classes of $M$. In particular, symplectic circle actions with no fixed points are never Hamiltonian. This leads to the following important question: What conditions force a symplectic action with fixed points to be Hamiltonian? Frankel proved that Kahler circle actions with fixed points on Kahler manifolds are always Hamiltonian. In contrast, McDuff constructed a non-Hamiltonian symplectic circle action with fixed tori. Despite significant additional research, the following question is still open: Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points? The main goal of this talk is to answer this question by constructing a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed six-dimensional symplectic manifold. In part, joint with Jordan Watts.

#### March 23, 2015: Ciprian Manolescu, "Seiberg-Witten Floer K-theory and its applications"

Abstract: In the mid 1990's Furuta proved his 10/8 theorem: If X is a closed spin four-manifold, then its second Betti number is at least 10/8 times the signature. The proof involves defining a finite dimensional approximation of the Seiberg-Witten map, and analyzing the induced map on equivariant K-theory. In this talk I will describe an extension of the 10/8 theorem to four-manifolds with boundary. The corresponding inequality involves a term derived from the Pin(2)-equivariant Seiberg-Witten Floer K-theory of the boundary. Furthermore, a slightly improved inequality (due to Jianfeng Lin) can be proved using KO-theory instead of K-theory.

#### March 25, 2015: Barney Bramham, "Floer-like complexes for surfaces, maximally unlinked braids, and finite energy foliations"

Abstract: I will talk about an approach to constructing finite energy foliations by pseudo-holomorphic curves with prescribed asymptotic orbits. The idea is that so called maximally unlinked braids of periodic orbits support a Floer-like chain complex. The concept of unlinkedness comes from LeCalvez work on surface homeomorphisms, as I will explain. The upshot is that it allows us to essentially characterize finite energy foliations for mapping tori: also, these chain complexes should be of independent interest. I will draw a lot of pictures.

#### March 27, 2015: Eugene Gorsky, "Torus knots and Cherednik algebras"

Abstract: I will describe the relation between the colored HOMFLY-PT invariants of torus knots and the characters of certain representations of rational Cherednik algebras. A conjectural lift of this relation to Khovanov-Rozansky homology will be also discussed. No knowledge of Cherednik algebras will be assumed, all necessary definitions will be given in the talk.

#### April 3, 2015: Roger Casals, "Overtwisted contact structures"

Abstract: In this talk we discuss a criterion that characterizes overtwisted contact structures. The result establishes seven equivalent properties that relate overtwistedness to loose Legendrians, adapted open books decompositions and thick neighborhoods of overtwisted submanifolds. We first provide the necessary definitions and properties involved in the criterion, and then present the proof of (part of) these equivalences. Finally, we explore possible consequences and applications. This is joint work with E. Murphy and F. Presas.

#### April 10, 2015, 9:30 a. m.: Sobhan Seyfaddini, "Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants."

Abstract: Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez’ theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by N, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.

#### April 10, 2015, 10:45 a. m. : Jake Rasmussen, "L-Space filling slopes of homology solid tori"

Abstract: A conjecture of Boyer, Gordon and Watson relates the property of being an L-space to the non-left-orderability of the fundamental group. A closely related problem, introduced by Boyer and Clay, is to take a 3-manifold with torus boundary and ask which of its Dehn fillings are L-spaces (and which are LO). I'll characterize the set of L-space filling slopes of a homology $S^1\times D^2$ in terms of its Alexander polynomial, and discuss some applications to questions raised by Boyer and Clay, and by Hanselman. If time permits, I may also describe some Seiberg-Witten theory motivation. Joint work with Sarah Rasmussen.

#### April 17, 2015: Chris Woodward, "Floer theory and the minimal model program"

Abstract: The Fukaya category is expected to be a categorification of the cohomology of a symplectic manifold in many cases. Despite this expectation, the Fukaya category is only known to be non-trivial in some special cases. I will discuss some partial results towards the following conjecture: If the manifold has a stacky-smooth running of the minimal model program then the Fukaya category is generated by Lagrangians associated to the minimal model transitions. For toric varieties, this is essentially a reformulation of a results of Abouzaid-Fukaya-Oh-Ohta-Ono. I will discuss the case of representation varieties of punctured two-spheres where a minimal model program and the associated Lagrangians can be described explicitly. This project uses a foundational system that is joint with Charest, and another part of the project is joint with Schultz.

#### April 24, 2015: Nate Bottman, "Quilts with figure eight singularity: progress report"

Abstract: In work-in-progress with Katrin Wehrheim, we aim to bind together the Fukaya categories of many different symplectic manifolds into a single algebraic object. This object is the "symplectic $A_{\infty}$-2-category", whose objects are symplectic manifolds, and where $\text{hom}(M, N) := Fuk(M^{−}\times N).$ At the core of our project are pseudoholomorphic quilts with figure eight singularity, and I will discuss recent progress: toward the construction of the moduli space of domains on one hand, and toward establishing the Fredholm property on the other.

#### May 1, 2015: Allison Moore, "Cosmetic crossing changes in thin knots"

Abstract: The cosmetic crossing conjecture asserts that the only crossing changes which preserve the oriented isotopy class of knot are nugatory. Currently, the knots that are known to satisfy this classic conjecture include two-bridge and fibered knots. We will show that Khovanov-thin knots (a class of knots which includes all alternating and quasi-alternating knots) also satisfy the cosmetic crossing conjecture, provided the first homology of the branched double cover decomposes into summands of square-free order. We will also discuss some generalizations. The proof relies on the Dehn surgery characterization of the unknot, a tool coming from Floer homology. This is joint work with Lidman.

## 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.