The SGGTC seminar meets on Fridays in Math 407, at 10:45 am unless noted otherwise (in red).
Our e-mail list.
Joint with Geometric Topology Seminar
|Instantons and odd Khovanov homology|
|Symmetric Self-adjoint Hopf categories and a categorical Heisenberg double|
|Symplectic fillings and star surgery|
Yajing Liu (UCLA)
|L-space surgeries on links|
Siu-Cheong Lau (Harvard)
|Generalized SYZ mirror symmetry|
Matt Hogancamp (Indiana)
|The stable Khovanov-Rozansky homology of torus knots|
Andrew Manion (Princeton)
|U_q(gl(1|1))-representations and Ozsvath-Szabo's new bordered theory|
| Oct. 24
Dan Cristofaro-Gardiner (Harvard)
IAS, Simonyi Hall 101.
| Symplectic embeddings from concave toric domains into convex ones
(Joint Columbia-Princeton-IAS Symplectic Topology Seminar)
IAS, Simonyi Hall 101.
| Beyond ECH capacities
(Joint Columbia-Princeton-IAS Symplectic Topology Seminar)
Lucas Culler (Princeton)
|3-manifolds modulo surgery triangles|
Khoa Nguyen (Stanford)
|On symplectic homology of complement of a normal crossing divisor|
|Trace of the categorified quantum groups|
|Stability in Fukaya categories of surfaces|
|The Lefschetz Hyperplane Theorem is Mostly Wrong (symplectically speaking)|
Erkao Bao (UCLA)
|Definition of cylindrical contact homology in dimension 3|
Tye Lidman (UT Austin)
|Contact topology and the Cabling conjecture|
No seminar -- Thanksgiving holiday
(University of Warsaw)
|Deformation of singular points of plane curves from smooth point of view|
|Exotic t-structures for two-block Springer fibers|
|Abelian Gauge Theory, Knots, and odd Khovanov Homology|
[First day of final exams]
Abstract: Consider the double cover of the 3-sphere branched over a link. We will discuss how the instanton homology of this 3-manifold is related to the Khovanov homology of the link. When signs (integer coefficients) are taken into account, the relevant link homology is odd Khovanov homology, which is genuinely distinct from Khovanov homology. This material is motivated by work of Ozsváth and Szabó in the Heegaard-Floer setting.
September 19, 2014: Lena Gal, "Symmetric Self-adjoint Hopf categories and a categorical Heisenberg double"
Abstract: We use the language of higher category theory to define what we call a
"symmetric self-adjoint Hopf" (SSH) structure on a semisimple abelian category. SSH categories are the categorical analog of positive self-adjoint Hopf algebras studied by A. Zelevinsky. It follows from his work that for every positive self-adjoint Hopf algebra the Heisenberg double is equipped with a natural action on the algebra. We obtain categorical analogs of the Heisenberg double and its action from the SSH structure on a category in a canonical way. We exhibit the SSH structure on the category of polynomial functors. The categorical Heisenberg double in this case provides a categorification of the infinite dimensional Heisenberg algebra related to the categorification proposed by M. Khovanov. The preprint is available on arXiv:1406.3973.
Abstract: Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations called star surgery to construct examples of exotic 4-manifolds.
Abstract: An L-space link is a link on which all large surgeries are L-spaces. These links turn out to be rich in geometry and simple in algebra. I will present some properties and examples of L-space links in contrast to L-space knots, give bounds on the ranks of their Floer homology and on the coefficients in the multi-variable Alexander polynomials, and show that how to use Manolescu-Ozsvath link surgery formula to give a free resolution of the Floer homology of surgeries on any L-space links. As a result, we compute the whole package of Heegaard Floer homology of surgeries on 2-component L-space links only in terms of Alexander polynomial and the surgery framing. This work is inspired by the work of Ozsvath-Szabo, Gorsky-Nemethi, and J.Hom.
Abstract: Mirror symmetry conjecture reveals deep relations between symplectic and complex geometries. The conjecture has been verified in many interesting cases by explicit computations. However the computations do not explain why we should expect mirror symmetry. In this talk I will introduce our constructive approach to mirror symmetry, which has the advantage that it always comes with a functor which transforms Lagrangian submanifolds to twisted holomorphic vector bundles (or known as matrix factorizations) over the mirror, and hence explains why symplectic geometry is mirror to complex geometry. This is a joint work with Cheol-Hyun Cho and Hansol Hong.
October 10, 2014, 10:45 a.m.: Matt Hogancamp, "The stable Khovanov-Rozansky homology of torus knots"
Abstract: In a series of surprising conjectures, Gorsky-Oblomkov-Rasmussen-Shende relate Khovanov-Rozansky homologies of the torus knot T(p,q) with the representation theory of the rational Cherednik algebra for SL_p. One such conjecture states that the triply graded Khovanov-Rozansky homology has a well-defined limit as $q\to \infty$, and that this homology is very simple: it is a (super) polynomial ring. In this talk we discuss and outline a proof of this limiting version of their conjecture. Along the way we discuss a categorified Young symmetrizer, which plays an essential role in our proof.
October 17, 2014: Andrew Manion, " U_q(gl(1|1))-representations and Ozsvath-Szabo's new bordered theory"
Abstract: I'll start with a brief overview of some recent constructions, due to Ozsvath and Szabo, of a bordered theory for knot Floer homology based on the Kauffman-states Heegaard diagram for a knot. Then I'll discuss the decategorification of this theory and how it relates to representations of U_q(gl(1|1)). Finally, I'll talk about a more precise connection between one type of local Ozsvath-Szabo bimodule and the bimodules over quiver algebras defined by Khovanov and Seidel.
October 24, 2014, 11:00am: Dan Cristofaro-Gardiner, " Symplectic embeddings from concave toric domains into convex ones"
Abstract: Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one ellipsoid into another. We explain why ECH capacities give a sharp obstruction to embedding any "concave toric domain" into a "convex" one. We also explain why the ECH capacities of any concave or convex toric domain are determined by the ECH capacities of a corresponding collection of balls. Some of this is joint work with Keon Choi, David Frenkel, Michael Hutchings, and Vinicius Ramos.
Abstract: ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain” (see previous talk). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. In this talk we explain how more refined information from ECH gives stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.
Abstract: Consider the abelian group $K_g$ generated by bordered 3-manifolds with boundary a genus g surface $\Sigma_g$, modulo the relation that any three manifolds which are related by a surgery triangle sum to zero. In this talk I'll prove that $K_g$ is free and describe an explicit finite basis. This basis is of interest because it generates the category assigned to $\Sigma_g$ by any bordered Floer theory with exact triangles.
Abstract: In this talk, we discuss our work in progress about how degeneration of the divisor at infinity into a normal crossing divisor affects the symplectic homology of an affine variety. From an anti-surgery picture, by developing an anti-surgery formula for symplectic homology similar to work by Bourgeois-Ekholm-Eliashberg, we show that essentially, the change in symplectic homology is reflected by the Hochschild invariants of the Fukaya category of a collection of Lagrangian spheres on the smooth divisor.
Abstract: We show that the center and the trace of any 2-representation of the Khovanov-Lauda 2-category for any symmetrizable Lie algebra g admit an action of the current algebra U(g[t]). In particular, the 0th Hochschild (co)homology of the cyclotomic quotient is the (dual) Weyl module.
Abstract: I will report on recent joint work with L. Katzarkov and M. Kontsevich (arXiv:1409.8611) in which we construct Bridgeland stability conditions on partially wrapped Fukaya categories of surfaces. Stable objects in these stability conditions correspond to finite-length geodesics (with local system) with respect to some flat metric with conical singularities given by a quadratic differential. The proof relies on the fact that all objects in the derived Fukaya category of a surface are geometric, i.e. come from immersed curves with local system. Examples and some further directions will also be discussed.
November 14, 2014, 2:30 p.m.: Mark McLean, "The Lefschetz Hyperplane Theorem is Mostly Wrong (symplectically speaking)"
Abstract: We show that for any symplectic manifold of dimension 2n>4, there exists a symplectic hypersurface Poincare dual to some positive multiple of the symplectic form whose (n-2)th Betti number is as large as we like. The idea here is to find an appropriate Liouville domain inside each of these symplectic manifolds and use Donaldson's asymptotically holomorphic techniques from his 1996 paper to find a symplectic hypersurface not intersecting some deformation of this Liouville domain.
November 21, 2014, 9:15 a.m.: Erkao Bao, " Definition of cylindrical contact homology in dimension 3"
Abstract: In this talk, we will give a rigorous definition of cylindrical contact homology for contact manifold of dimension 3, when there is no contractible Reeb orbit, and prove that it is an invariant of the contact structure. The proof is essentially based on some type of obstruction bundle calculations. Also, if time permits, we will talk about the case when there are contractible Reeb orbits. This talk is based on a joint work of Ko Honda and myself.
Abstract: The Cabling conjecture is a classical problem in three-manifold topology which attempts to characterize the knots in the three-sphere for which Dehn surgery results in a non-prime manifold. We use standard results in three- and four-dimensional contact and symplectic topology to study this problem. This is joint work with Steven Sivek.
December 3, 2014, 9:00 a.m.: Maciej Borodzik, "Deformation of singular points of plane curves from smooth point of view"
Abstract: Using Heegaard Floer theory we reprove a recent result by Gorsky and Nemethi about semicontinuity of semigroups under deformations of singular points. Unlike the original result, we add an extra assumption namely that the deformation is delta-constant, however the proof is purely topological. We explain the geometric meaning of this extra assumption and show a generalization. The generalization suggests a possible difference between smooth concordance and analytic deformations. This is a joint work with Charles Livingston.
Abstract: We study the exotic t-structure on D_n, the derived category of coherent sheaves on the Springer fiber for a two-block nilpotent (i.e. for a nilpotent matrix of type (m+n,n) in type A). Exotic t-structures were introduced by Bezrukavnikov and Mirkovic in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors between these categories D_n, indexed by affine tangles. We use these functors to enumerate the irreducibles in the heart of D_n by crossingless matchings, and compute the Ext's between these irreducibles. This is joint with R. Anno
December 5, 2014, 10:45 a.m.: Aliakbar Daemi, "Abelian Gauge Theory, Knots, and odd Khovanov Homology"
Abstract: In the first part of the talk I'll explain how one can use abelian gauge theory and Floer homological techniques to define a (3+1)-dimensional "Topological Quantum Filed Theory". In the second talk some relations between this invariant and odd Khovanov homology will be discussed. No prior knowledge of gauge theory, Floer homology, or odd Khovanov homology will be expected.