|Sept. 5, 2:30 p.m.||Organizational meeting.|
|Aaron Lauda (Columbia)||Categorification of Quantum Groups|
|Sept. 19||Stephan Wehrli (Columbia)||A relationship between reduced colored Khovanov homology and knot Floer homology|
|Sept. 26||Krzysztof Putyra (Jagiellonian U., Krakow)||Cobordisms with chronology and a functorial description of odd link homology|
|Oct. 3||Marco MacKaay (U. do Algarve)||The 1,2-coloured HOMFLY-PT link homology|
|Oct. 10||Frol Zapolsky (Tel Aviv U.)||Symplectic quasi-states and applications|
|Oct. 17||C-C. Melissa Liu (Columbia)||T-duality and equivariant homological mirror symmetry for toric varieties|
|1:00: Aleksey Zinger (SUNY Stonybrook)||From Gromov-Witten invariants to integer counts|
|3:30: Mohammed Abouzaid (MIT)||A restriction functor in Lagrangian Floer Homology|
|Oct. 31||Florent Schaffhauser (Keio U.)||Decomposable representations of surface groups.|
|Nov. 7||1:00: Liam Watson (UQÀM and CIRGET, Canada)||Involutions on 3-manifolds and Khovanov homology|
|3:30: Vera Vertesi (Eötvös Loránd U., Hungary)||Legendrian knots and Heegaard Floer homologies|
|Chris Woodward (Rutgers)||Gauged Gromov-Witten theory and morphisms of CohFT's|
|1:00: Cagatay Kutluhan (U. Michigan)||Seiberg-Witten Floer homology and symplectic forms on S^1xM^3|
|3:30: Michael Hutchings (U.C. Berkeley)||The Weinstein conjecture for stable Hamiltonian structures|
|Nov. 28||No seminar this week.|
|Dec. 5||Michael Chance (SUNY Stony Brook)||TBA|
Adam Lowrance (Louisiana State U.)
|Cube diagrams and a homology theory for knots|
Abstract: Crane and Frenkel proposed that 4-dimensional TQFTs could be obtained by categorifying quantum groups at root of unity using their canonical bases. In my talk I will explain how the quantum enveloping algebra of quantum sl(2) at generic q can be categorified using a diagrammatic calculus. If time permits I will also explain joint work with Mikhail Khovanov on how this construction can be generalized to quantum sl(n). No background on quantum groups will be assumed.
Abstract: This is joint work with Elisenda Grigsby.
For an (n,n)-tangle T in the ball B^3, we describe a spectral sequence whose E^2 term is a suitable variant of Khovanov's homology of T, and which converges to the sutured Floer homology of the branched double-cover of B^3, branched along T. As an application, we show that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot for all n>1.
Abstract: I will enrich cobordisms with special projections on the closed interval I=[0; 1] to break the symmetry of oriented cobordisms. This creates a new category, which in case of dimension two has a nice presentation by generators and relations. This category can be used to give a functorial description of the construction of odd link homology as well as to define a new type of TQFT's.
I will build the Khovanov complex Kh(T) for a given tangle diagram in the category of cobordisms with chronology. It is invariant under Reidemeister moves up to chain homotopies, relations analogous to Bar-Natan's S/T/4Tu and a condition given by a chronology change. Then any chronological TQFT satisfying these additional conditions defines a complex in the category of modules and we can compute its homology. This procedure generalises both Khovanov and odd link homology theories.
Abstract: This is joint work with Marko Stosic and Pedro Vaz. In my talk I will show how to generalize the triply graded HOMFLY-PT link homology, due to Khovanov and Rozansky, to links with components labelled 1 and 2. The approach we follow uses bimodules and Hochschild homology, using ideas inniated by Khovanov for ordinary HOMFLY-PT link homology.
Abstract: Symplectic quasi-states are certain functionals which arose at the intersection of symplectic topology and (modern) functional analysis. The specific symplectic quasi-states which will be defined in this talk come from Floer homology and consequently enjoy many nice properties. These properties allow for various applications, and we'll demonstrate two of them: restrictions on partitions of unity and simultaneous measurements in classical mechanics.
Abstract: Homological mirror symmetry for toric varieties has been studied extensively in recent years. In this talk, I will discuss equivariant homological mirror symmetry for toric varieties based on a joint work with Bohan Fang, David Treumann, and Eric Zaslow. The main ingredients in our argument are (i) the SYZ transformation, and (ii) the microlocalization functor relating the Fukaya category of the cotangent to construcible sheaves on a base.
New York Area Joint Symplectic Geometry Seminar (PDF)
Aleksey Zinger, "From Gromov-Witten invariants to integer counts" (iCal)
Abstract: Gromov-Witten invariants of a compact symplectic manifold are certain virtual counts of J-holomorphic curves. These rational numbers are rarely integer, but are generally believed to be related to some integer counts. In string theory, these counts are known as instaton numbers and BPS states. The predictions of Aspinwall-Morrison and Gopakumar-Vafa for the existence of BPS states of Calabi-Yau 3-folds are extended by Pandharipande to all 3-folds, by Klemm-Pandharipande to all Calabi-Yau varieties in genus 0 and Calabi-Yau 4-folds in genus 1, and by Pandharipande and the speaker to Calabi-Yau 5-folds in genus 1. The last extension came as a bit of a surprise to some string theorists, who also feel that extensions to higher dimensions are impossible. The aim of this talk is to survey the known predictions, indicating how they arise, how the 6-dimensional case differs from low-dimensional cases, and why they hold for Fano classes in 3-folds (symplectic manifolds of real dimension 6).
Mohammed Abouzaid, "A restriction functor in Lagrangian Floer Homology" (iCal)
Abstract: Consider an exact Lagrangian submanifold of an exact symplectic manifold (say an affine variety). Using a model for the (wrapped) Fukaya category of a cotangent bundle as modules over chains on the based loop space, I will define a restriction functor from the Fukaya category of the ambient symplectic manifold to the (wrapped) Fukaya category of the cotangent bundle. Time permitting, I will outline a conjectural version for plumbings.
Abstract: In this talk, we generalize to arbitrary surface groups and arbitrary compact connected Lie groups the notion of decomposable representation, first introduced by Falbel and Wentworth for unitary representations of the punctured sphere group. We show that such decomposable representations are the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space of representations, forming therefore a Lagrangian submanifold of this moduli space. The existence of decomposable representations is obtained as a corollary of a real convexity theorem for group-valued momentum maps.
Abstract: Given a manifold with torus boundary, together with an appropriate involution, it is possible give obstructions to certain exceptional surgeries using Khovanov homology. In particular, obstructions to lens space surgeries, as well as obstructions to finite fillings may be obtained. This talk will explain how these obstructions arise, and attempt to compare them with strong obstructions arising in Heegaard-Floer homology.
Abstract: Using the language of Heegaard Floer homology recently two different invariants were defined for Legendrian and transverse knots in a contact 3-manifold. Both of them arises from the generalization of the contact invariant in Heegaard Floer homology. The Legendrian invariant defined by Lisca, Ozsváth, Stipsicz and Szabó takes its values in knot Floer homology, while the other one is in the sutured Floer homology, defined as the EH-class of Honda, Kazez and Matic for the complement of a Legendrian knot. In this talk I will give a brief description of both of these invariants, and describe their relation. As a corollary we will obtain, that the Legendrian invariant vanishes for knots having Giroux-torsion in their complement. This is joint work with András Stipsicz.
New York Area Joint Symplectic Geometry Seminar (PDF)
Cagatay Kutluhan, "Seiberg-Witten Floer homology and symplectic forms on S^1xM^3" (iCal)
Abstract: Let M be a closed, connected, orientable 3-manifold. Subject to a monotonicity condition, we calculate the Seiberg-Witten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we show that M fibers over the circle if it has first Betti number 1 and S^1 X M admits a symplectic form with non-torsion canonical class. This is joint work with Cliff Taubes.
Michael Hutchings, "The Weinstein conjecture for stable Hamiltonian structures" (iCal)
Abstract: We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits. Joint work with Cliff Taubes.
Abstract: A cube diagram is a representation of a knot embedding that is closely related to grid diagrams. I will present a finite set of moves that generate all isotopies of cube diagrams. Finally, I will describe a homology theory based upon cube diagrams. This homology theory is completely determined by the knot Floer homology of the knot.
Other relevant information.
- Columbia-NYU-Stony Brook Joint Symplectic Geometry Seminar.
- Columbia's Informal Symplectic Geometry Seminar. (Currently in hiatus.)
- Columbia Geometric Topology Seminar.
- Columbia Algebraic Geometry Seminar.
- Courant Symplectic Geometry Seminar.
- SUNY Stony Brook Symplectic Geometry Seminar.
- Princeton Topology Seminar.
- Princeton Symplectic Geometry Seminar.
- Stay tuned!
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. You can subscribe directly via Google Groups or by contacting R. Lipshitz.