0;269;0c Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar

The SGGTC seminar meets on Fridays in Math 417, at 10:45 am unless noted otherwise (in red).

Previous semesters: Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.

Future semesters: Spring 2012

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(iCal)
SGGTC Seminar Schedule

Date Speaker Title
Sep 9
10:45 am		 Cheol-Hyun Cho
Seoul National University/
Northwestern University		 Lagrangian Floer cohomology for toric orbifolds
Sep 9
1:20 pm
Math 520		 Organizational Meeting
 Organizational Meeting
Joint with Geometric Topology Seminar
Sep 16 No seminar this week
Sep 23 Nicolo Sibilla
Northwestern University		 Mirror Symmetry and Ribbon Graphs
Sep 30 Anthony Licata
IAS/Australian National University		 Affine Lie algebras, Hilbert schemes and Categorification
Oct 7 Egor Shelukhin
Tel Aviv University		 Action homomorphisms, moment maps and quasimorphisms
Oct 14 Maksim Lipyanskiy
Columbia University		 Gromov-Uhlenbeck Compactness

Oct 21
9:30 am

 Emmanuel Wagner
Institut Mathématique de Bourgogne		 Alexander polynomial and enhanced Kauffman states
Oct 21
10:45 am		 Sobhan Seyfaddini
UC Berkeley		 C^0 limits of Hamiltonian paths and spectral invariants
Oct 28 Anton Zeitlin
Yale University		 Lian-Zuckerman homotopy algebras, Courant/Vertex algebroids and beta-functions of string theory
Nov 4
10:45 am		 Florent Schaffhauser
University of Los Andes		 Yang-Mills equations over Klein surfaces
Nov 4
3:30 pm
Math 507		 Raphael Rouquier
UCLA		 Kazhdan-Lusztig theory and Calogero-Moser spaces
Nov 11 Steven Sivek
Harvard University		 Monopole Floer homology and Legendrian knots
Nov 18 Rumen Zarev
UC Berkeley
 Toward a minus version of bordered Heegaard Floer homology
Nov 25 No seminar this week
Dec 1
2:00 pm
Math 622		 Dustin Ross
Colorado State University		 Open GW Theory and the Orbifold Vertex
Dec 2 Yang Huang
USC		 An absolute grading of Heegaard Floer homology by homotopy classes of 2-plane fields
Dec 9 John Baldwin
Princeton University		 The equivalence of transverse link invariants in knot Floer homology

Dec 16
9:30 am

 Nick Sheridan
MIT		 Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space
Dec 16
10:45 am		 Eugene Gorski
Stony Brook University		 A model for the stable Khovanov homology of torus knots

 

Abstracts

September 9, 2011

Cheol-Hyun Cho, "Lagrangian Floer cohomology for toric orbifolds"

Abstract: Floer cohomology for Lagrangian torus fibers in toric manifolds has been extensively studied in the last decade. We discuss an analogous theory for toric orbifolds by studying holomorphic orbifold discs. We classify such orbi-discs in Fano toric orbifolds, compute Floer cohomology and discuss non-displaceability of Lagrangian torus fibers in these cases.

September 23, 2011

Nicolo Sibilla, "Mirror Symmetry and Ribbon Graphs"

Abstract: In this talk, I will explain how to construct a model for the Fukaya category of punctured Riemann surfaces in terms of a sheaf of dg-categories over a suitable category of ribbon graphs. Using this model, I will be able to prove a homological mirror symmetry statement for degenerate ellitpic curves. This is joint with Treumann and Zaslow. If time permits, I will also discuss recent work of myself on mapping class group actions on derived categories of coherent sheaves, which is motivated by the above mirror symmetry framework.

September 30, 2011

Anthony Licata, "Affine Lie algebras, Hilbert schemes and Categorification"

Abstract: The basic representation of a simply-laced affine Kac-Moody Lie algebra was constructed by Frenkel-Kac and Segal using "vertex operators". We describe a categorification of this construction and explain how this categorification appears in the geometry of Hilbert schemes of points on the resolution of a simple singularity. (Joint with Sabin Cautis)

October 7, 2011

Egor Shelukhin, "Action homomorphisms, moment maps and quasimorphisms"

Abstract: Whenever a group acts on a symplectic space with an equivariant moment map, an analogue of Weinstein's action homomorphism can be defined. If there is also an invariant system of 'geodesic' paths, such that the corresponding geodesic triangles have uniformly bounded symplectic areas, we define a quasimorphism on the universal cover of the group. This applies to the action of the Hamiltonian group on the space of compatible almost complex structures (with the moment map of Donaldson and Fujiki) producing a non-trivial quasimorphism on the universal cover of Ham for every symplectic manifold of finite volume. We discuss this construction, some additional properties and applications.

October 14, 2011

Maksim Lipyanskiy, "Gromov-Uhlenbeck Compactness"

Abstract: We introduce an analytic framework that, in special circumstances, unites Yang-Mills theory and the theory of pseudoholomorphic curves. As an application of these ideas, we discuss the relation between the categorification of the Casson invariant based on gauge theory and symplectic geometry.

October 21, 2011, 9:30 a.m.

Emmanuel Wagner, "Alexander polynomial and enhanced Kauffman states"

Abstract: State sum formulas play a key role in quantum topology, and are even more relevant for the new purpose of categorification. Among polynomial link invariants, the two most popular are probably the Alexander polynomial and the Jones polynomial. The Jones polynomial has a natural state sum formula in terms of so called enhanced Kauffman states. In this talk, we will see that the set of enhanced Kauffman states can also be used as a set of states for a formula for the Alexander polynomial.

October 21, 2011, 10:45 a.m.

Sobhan Seyfaddini, "C^0 limits of Hamiltonian paths and spectral invariants"

Abstract: After briefly reviewing spectral invariants, I will write down an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^0-distance of its flow from the identity. I will also show that, unlike the Hofer norm, the spectral norm is C^0-continuous on surfaces. Time permitting, I will present an application to the study of area preserving disk maps.

October 28, 2011

Anton Zeitlin, "Lian-Zuckerman homotopy algebras, Courant/Vertex algebroids and beta-functions of string theory"

Abstract: Homotopy algebras of Lian and Zuckerman for topological vertex operator algebras (TVOA) will be discussed. In a paticular case of semi-infinite complex of Virasoro algebra, it appears that the corresponding Maurer-Cartan equations reproduce beta-functions of sigma models for certain VOA, i.e. Yang-Mills and Einstein equations with stringy corrections. As a byproduct this leads to a surprising relation between Courant algebroids and classical field equations.

November 4, 2011, 10:45 a.m.

Florent Schaffhauser, "Yang-Mills equations over Klein surfaces"

Abstract: Moduli spaces of semi-stable algebraic vector bundles on a real algebraic curve / Klein surface can be constructed using a gauge-theoretic approach, in which they are viewed as spaces of solutions to Yang-Mills equations over a compact Riemann surface, satisfying an additional Galois symmetry. From the symplectic point of view, these moduli spaces are Lagrangian quotients, defined by means of an involution on the space of all unitary connections on a fixed Hermitian bundle. By adapting the equivariant approach of Atiyah and Bott to a setting with involutions, we compute the mod 2 Poincare polynomial of these Lagrangian quotients. This is joint work with Chiu-Chu Melissa Liu.

November 4, 2011, 3:30 p.m.

Raphael Rouquier, "Kazhdan-Lusztig theory and Calogero-Moser spaces"

Abstract: Rational double affine Hecke algebras are related to finite-dimensional Hecke algebras via monodromy representations. By degenerating both sides, one expects a relation between Calogero-Moser spaces and Lusztig's asymptotic ring. We will explain a conjectural construction of left cells in Weyl groups in terms of ramification for Calogero-Moser spaces. (joint work with Cedric Bonnafe)

November 11, 2011, 10:45 a.m.

Steven Sivek, "Monopole Floer homology and Legendrian knots"

Abstract: We will define invariants of Legendrian knots using Kronheimer and Mrowka's construction of monopole Floer homology for sutured manifolds. We will discuss several interesting properties of these invariants, including behavior under stabilization and contact surgery which suggests that they are closely related to the Lisca-Ozsvath-Stipsicz-Szabo invariant in knot Floer homology, and show that they are functorial with respect to Lagrangian concordance.

November 18, 2011, 10:45 a.m.

Rumen Zarev, "Toward a minus version of bordered Heegaard Floer homology"

Abstract: Bordered Heegaard Floer homology, in its current state, can recover the hat version of Heegaard Floer homology for closed manifolds. It is related to the sutured Floer homology, and gluing sutured manifolds along surfaces with boundary. One idea for expanding the bordered framework to obtain the stronger plus/minus versions of HF, is to look at sutured manifolds and gluing along closed boundary components. I will discuss progress along this lines, as well as connections to other possible approaches.

December 1, 2011, 2:00 p.m.

Dustin Ross, "Open GW Theory and the Orbifold Vertex"

Abstract: Almost 10 years ago, Aganagic, Klemm, Marino, and Vafa proposed the topological vertex, a basic building block for Gromov-Witten theory in all genera of toric Calabi-Yau 3-folds. The topological vertex has now been realized in a number of equivalent ways: via Schur functions, 3d partitions with prescribed asymptotics, and three-partition Hodge integrals on the moduli space of curves. The orbifoldvertex is the natural extension of the topological vertex to toric orbifold targets. I will discuss recent work in which the orbifold vertex is realized as a generating function of open orbifold GW invariants of the orbifold [C^3/G], defined by generalizing the open invariants of Katz and Liu. I will also discuss the conjectural correspondence with the DT orbifold vertex developed by Bryan-Cadman-Young.

December 2, 2011, 10:45 a.m.

Yang Huang, "An absolute grading of Heegaard Floer homology by homotopy classes of 2-plane fields"

Abstract: In 2001, P. Ozsvath and Z. Szabo introduced Heegaard Floer homology, which is an invariant of closed, oriented 3-manifolds. The Heegaard Floer homology group splits into a direct sum according to Spin^c structures. Each direct summand is relatively graded by Z mod d where d is the divisibility of the first Chern class of the Spin^c structure. In this talk we discuss an absolute grading of Heegaard Floer homology by homotopy classes of 2-plane fields in a 3-manifold. In particular we define an absolute Q-grading for torsion Spin^c structures which coincides with the one defined by Ozsvath-Szabo. This is joint work with Vinicius Gripp.

December 9, 2011, 10:45 a.m.

John Baldwin, "The equivalence of transverse link invariants in knot Floer homology"

Abstract: Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston dened an invariant of transverse knots in the tight contact 3-sphere. A bit later, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they overlap. I'll discuss a proof of this conjecture. Our strategy is to define yet another invariant of transverse knots and show that this third invariant agrees with the two invariants mentioned above. This is joint work with Vera Vertesi and David Shea Vela-Vick.

December 16, 2011, 9:30 a.m.

Nick Sheridan, "Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space"

Abstract: We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold (also considered by Nohara-Ueda, using our work). After stating the result carefully, we will describe some of the techniques used in its proof, and draw pictures in the one-dimensional case.

December 16, 2011, 10:45 a.m.

Eugene Gorsky, "A model for the stable Khovanov homology of torus knots"

Abstract: In 2005 S. Gukov, N. Dunfield and J. Rasmussen conjectured that the Khovanov homology of some knots can be computed by taking the homology of a certain differential acting on triply graded homology of a knot. They also conjectured the simple description of the stable limit of the triply graded homology for (n,m) torus knots when m is large. I will discuss the conjectural algebraic description of the differential and its homology that matches all known experimental data. When both m and n tend to infinity, the Poincare series for the homology turns out to be related to the Rogers-Ramanujan identity. The talk is based on a joint work with A. Oblomkov and J. Rasmussen.

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