The SGGTC seminar meets on Fridays in Math 417, at 10:45 am unless noted otherwise.
Next semester: Spring 2011
Unusual time this week only: 1:15 pm.
|Sep 17||Ben Elias
|A diagrammatic categorification of the Hecke algebra|
|Sep 24||Adam Knapp
|Monopoles, cobordisms, and exact triangles|
|Oct 1||Florent Schaffhauser
(University of Los Andes)
|Moduli of real and quaternionic bundles over a curve|
|Oct 8||Adam Levine
|Twisted coefficients and the unoriented skein sequence for HFK|
|Oct 15||Yiqiang Li
|Geometric realizations of quantum groups|
|Oct 22||Sikimeti Ma'u
|Quilts and A-infinity structures|
|Oct 29||Michael Usher
(University of Georgia)
|Deformed Hamiltonian Floer theory and Calabi quasimorphisms|
|Nov 5||Allison Gilmore
|An algebraic proof of invariance for knot Floer homology|
|Note: There are two talks on November 12!|
|Heegaard Floer meets Seiberg--Witten|
|Jeremy Van Horn-Morris
(American Institute of Mathematics)
|Spinal open book decompositions and symplectic fillings|
|Nov 19||Eamonn Tweedy
|On the anti-diagonal filtration for the Heegaard Floer chain complex of a branched double-cover|
|SO(3) Kauffman Homology for Graphs and Links|
|Symplectic embeddings of polydisks|
4:10 pm - 5:00 pm
(University of Chicago)
|Symplectic Reduction of Quasi-morphisms and Quasi-states|
|A colored Khovanov bicomplex|
|Note: There are two talks on December 10!|
|Holomorphic Pairs of Pants in Mapping Tori|
|The Seiberg-Witten Equations on Manifolds with Boundary|
|Some analytic aspects of Vafa-Witten twisted N=4 supersymmetric Yang-Mills theory|
Unusual time this week only: 1:15 pm.
Ben Elias, "A diagrammatic categorification of the Hecke algebra"
Abstract: The Hecke algebra in Type A is a ubiquitous algebra in representation theory, knot theory, and geometry. Soergel provided a simple and easy-to-use categorification of the Hecke algebra, using bimodules over a polynomial ring. We explain this categorification, and briefly motivate it in the context of equivariant cohomology of flag varieties. Then, we give an even simpler version of the same categorification, using planar graphs to represent morphisms in Soergel's category (joint work w/ M. Khovanov). Finally, we will discuss applications to knot theory, including the functoriality of Khovanov-Rozansky link homology (joint work w/ D. Krasner).
Adam Knapp, "Monopoles, cobordisms, and exact triangles"
Abstract: Suppose W_0 is a cobordism between two 3-manifolds Y and Y_0. When Y_0 is a member of a certain class of surgery exact triangle (with 3-manifolds Y_0, Y_1, and Y_2), I provide a method of computing the map on Monopole Floer homology induced by W_0 in terms of the maps induced by cobordisms W_1 and W_2 from Y to Y_1 and Y_2 respectively.
Florent Schaffhauser, "Moduli of real and quaternionic bundles over a curve"
Abstract: We examine the moduli problem for real and quaternionic vector bundles over a curve, and we give a gauge-theoretic construction of moduli varieties for such bundles. These moduli varieties are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study the induced $Gal(C/R)$-action on moduli varieties of semistable holomorphic bundles over a complex curve with given real structure $\sigma$. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by $2^g + 1$, where $g$ is the genus of the curve. We show, moreover, that any two such connected components are homeomorphic.
Adam Levine, "Twisted coefficients and the unoriented skein sequence for HFK"
Abstract: We shall describe some work in progress with John Baldwin concerning Manolescu's unoriented skein exact sequence for knot Floer homology. Under the right conditions, this sequence can be iterated to give a cube of resolutions that computes HFK. Using twisted coefficients in a Novikov ring greatly simplifies this cube complex, since the homology of any resolution with multiple components vanishes. It is hoped that this approach may yield a new way to compute HFK combinatorially and shed some light its possible relation to Khovanov homology and the Heegaard Floer homology of the double branched cover.
Speaker: Yiqiang Li, "Geometric realizations of quantum groups"
Abstract: I'll recall Beilinson, Lusztig and MacPherson's classical work on the geometric realization of quantum groups of type A by double partial flag varieties. Then I'll present my recent work on the geometric realization of quantum groups of symmetric type by using localized equivariant derived categories of double framed representation varieties associated with a quiver.
Sikimeti Ma'u, "Quilts and A-infinity structures"
Abstract: I'll describe some A-infinity structures associated to Lagrangians and Lagrangian correspondences, particularly A-infinity modules, bimodules, and higher generalizations called n-modules. The structures can all be described pictorially in terms of quilted strips with markings, which are types of graph associahedra in disguise. The quilted strips are domains for holomorphic quilts (a la Wehrheim-Woodward), so they translate into A-infinity structure on the target symplectic manifolds. So, for example, a sequence of Lagrangian correspondences between symplectic manifolds M and N determines a bimodule of the Fukaya categories of M and N. More generally there is an A-infinity functor from Fuk(MxN) to Bimod(Fuk(M), Fuk(N)), which is really due to the underlying 3-module.
Michael Usher, "Deformed Hamiltonian Floer theory and Calabi quasimorphisms"
Abstract: I'll introduce a family of deformations of the Hamiltonian Floer complex on a symplectic manifold which, on passing to homology, recover the "big" quantum homology of the manifold. Using these deformations, one can construct Calabi quasimorphisms on the universal covers of the Hamiltonian diffeomorphism groups of new families of symplectic manifolds, including all one-point blowups.
Allison Gilmore, "An algebraic proof of invariance for knot Floer homology"
Abstract: We investigate the algebraic structure of knot Floer homology in the context of categorification. Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. Using this construction, we give a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. We close with an alternative description of knot Floer homology in terms of Soergel bimodules that suggests a close relationship with HOMFLY-PT homology.
Cagatay Kutluhan, "Heegaard Floer meets Seiberg--Witten"
Abstract: Recently Yi-Jen Lee, Clifford H. Taubes, and I have announced a proof of the conjectured equivalence between Heegaard Floer and Seiberg--Witten Floer homology groups of a closed, oriented 3-manifold. In this talk, I will try to outline our construction of this equivalence.
Jeremy Van Horn-Morris, "Spinal open book decompositions and symplectic fillings"
Abstract: Recently, C. Wendl used holomorphic curves to show that any strong filling of a planar open book comes as an extension of the open book to a Lefschetz fibration of the filling. I'll discuss a generalization of this result to spinal open books and discuss some applications, including a determination of fillability for contact structures on circle bundles. This is joint with S. Lisi and C. Wendl.
Eamonn Tweedy, "On the anti-diagonal filtration for the Heegaard Floer chain complex of a branched double-cover"
Abstract: One can use a grading from Seidel and Smith's fixed-point symplectic Khovanov cochain complex to obtain a filtration on the Heegaard Floer-hat chain complex for the two-fold cover of S^3 branched over a knot K (via an identification of generators). Although the definition comes from a braid whose closure is K, we in fact have that the resulting filtered chain homotopy type of the CF-hat complex is a knot invariant (as are the higher pages of the spectral sequence induced by this filtration). Under certain spectral sequence degeneration conditions (satisfied by all two-bridge knots, for instance), one obtains an absolute Maslov grading on HF-hat of the branched double-cover. We'll outline the definitions, discuss invariance, give some results, and make some further speculations related to this filtration.
Matt Hogancamp, "SO(3) Kauffman Homology for Graphs and Links"
Abstract: (Joint work with B. Cooper and S. Krushkal) There is a well-known relationship between the SO(3) Kauffman polynomial for links, the chromatic polynomial for planar graphs, and the 2-colored Jones polynomial. In this talk I will describe a categorification of this relationship using a categorified Jones-Wenzl projector on two strands, living in Bar-Natan's category. Some elementary properties will be discussed, as well as future directions.
Richard Hind, "Symplectic embeddings of polydisks"
Abstract: I will discuss some work in progress with Sam Lisi investigating obstructions to symplectically embedding four-dimensional polydisks into balls.
Matthew Borman, "Symplectic Reduction of Quasi-morphisms and Quasi-states"
Abstract: For certain complex hypersurfaces N in a Kahler manifold M, I will present a construction for symplectically reducing quasi-morphisms on the universal cover of the Hamiltonian group of M to quasi-morphisms on universal cover of the Hamiltonian group of N. Along the way I will show that spectral quasi-morphisms are the Calabi homomorphism on stably displaceable sets.
Noboru Ito, "A colored Khovanov bicomplex"
Andrew Cotton-Clay, "Holomorphic Pairs of Pants in Mapping Tori"
Abstract: We consider invariants of mapping tori of symplectomorphisms of a symplectic surface S, such as the symplectic field theory, contact homology, or periodic Floer homology for the standard stable Hamiltonian structure on the mapping torus. These invariants involve counts of holomorphic curves in R times the mapping torus. We obtain a number theoretic description of all rigid holomorphic curves in the case S = T^2, and obtain various pair-of-pants invariants for symplectomorphisms on higher genus surfaces. Our method involves reinterpreting counts of holomorphic pairs of pants in R times the mapping torus as counts of index -1 triangles between Lagrangians in S x S for certain 1-parameter families of almost-complex structures.
Timothy Nguyen, "The Seiberg-Witten Equations on Manifolds with Boundary"
Abstract: The analysis of the Seiberg-Witten equations have led to many important results in low-dimensional topology. These include the invariants defined by Witten for 4-manifolds and the monopole Floer invariants for 3-manifolds defined by Kronheimer-Mrowka and others. In both these situations, the equations and their moduli space of solutions are studied on closed manifolds. In this talk, we study the analysis of the Seiberg-Witten equations on manifolds with boundary. First, we discuss the space of solutions to the Seiberg-Witten equations on 3-manifolds with boundary. This solution space is infinite dimensional (even modulo gauge) since no boundary conditions are imposed on the equations. Second, we discuss how this solution space yields natural boundary conditions for the Seiberg-Witten equations on a cylindrical 4-manifold R x Y, where Y is a 3-manifold with boundary. We explain how the resulting nonlinear boundary value problem has well-posedness and compactness properties, and how these results therefore serve as foundational analysis for an eventual construction of a monopole Floer theory on manifolds with boundary.
Ben Mares, "Some analytic aspects of Vafa-Witten twisted N=4 supersymmetric Yang-Mills theory"
Abstract: Given an oriented Riemannian four-manifold equipped with a principal bundle, we investigate the moduli space of solutions to the Vafa-Witten equations. We establish various properties, computations, and estimates for these equations, and give a partial Uhlenbeck compactification of the moduli space.
Other relevant information
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.