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

Next semester: Spring 2011

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

Other area seminars. Our e-mail list.

Special Event: Columbia will host Categorification on Broadway, a graduate student Workshop on Tuesday, December 14 and Wednesday, December 15.

 Date Speaker Title Sep 101:15 pm Organizational Meeting Unusual time this week only: 1:15 pm. Sep 17 Ben Elias(Columbia) A diagrammatic categorification of the Hecke algebra Sep 24 Adam Knapp(Columbia) 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(Brandeis) Twisted coefficients and the unoriented skein sequence for HFK Oct 15 Yiqiang Li(Virginia Tech) Geometric realizations of quantum groups Oct 22 Sikimeti Ma'u(Columbia) Quilts and A-infinity structures Oct 29 Michael Usher(University of Georgia) Deformed Hamiltonian Floer theory and Calabi quasimorphisms Nov 5 Allison Gilmore(Columbia) An algebraic proof of invariance for knot Floer homology Note: There are two talks on November 12! Nov 129:30 am Cagatay Kutluhan(Columbia) Heegaard Floer meets Seiberg--Witten Nov 1210:45 am Jeremy Van Horn-Morris(American Institute of Mathematics) Spinal open book decompositions and symplectic fillings Nov 19 Eamonn Tweedy(UCLA) On the anti-diagonal filtration for the Heegaard Floer chain complex of a branched double-cover MONDAY Nov 22 4:00 pmMath 507 Matthew Hogancamp(UVA) SO(3) Kauffman Homology for Graphs and Links TUESDAYNov 304:45 pmMath 507 Richard Hind(Notre Dame) Symplectic embeddings of polydisks WEDNESDAYDec 14:10 pm - 5:00 pmMath 507 Matthew Borman(University of Chicago) Symplectic Reduction of Quasi-morphisms and Quasi-states MONDAYDec 64:00 pm Noboru Ito(Waseda University) A colored Khovanov bicomplex Note: There are two talks on December 10! Dec 109:30 am Andrew Cotton-Clay(Harvard) Holomorphic Pairs of Pants in Mapping Tori Dec 1010:45 am Timothy Nguyen(MIT) The Seiberg-Witten Equations on Manifolds with Boundary Dec 17Math 307 Ben Mares(McMaster University) Some analytic aspects of Vafa-Witten twisted N=4 supersymmetric Yang-Mills theory

# Abstracts

#### September 10, 2010

Organizational Meeting

Unusual time this week only: 1:15 pm.

#### September 17, 2010

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).

#### September 24, 2010

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.

#### October 1, 2010

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.

#### October 8, 2010

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.

#### October 15, 2010

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.

#### October 22, 2010

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.

#### October 29, 2010

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.

#### November 5, 2010

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.

#### November 12, 2010, 9:30 am

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.

#### November 12, 2010, 10:45 am

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.

#### November 19, 2010

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.

#### MONDAY, November 22, 2010, 4:00 pm in Math 507

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.

#### TUESDAY, November 30, 2010, 4:45 pm in Math 507

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.

#### WEDNESDAY, December 1, 2010, 4:10 pm - 5:00 pm

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.

#### MONDAY, December 6, 2010, 4:00 pm

Noboru Ito, "A colored Khovanov bicomplex"

#### December 10, 2010, 9:30 am

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.

#### December 10, 2010, 10:45 am

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.

#### December 17, 2010

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

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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.