Note: SGGT Seminar now meets in Math 520.

Next semester's web page.

The SGGT seminar meets on Fridays in Math 520 at 1:00 p.m.

Previous semesters: fall 2007, spring 2007, fall 2006.

Other area seminars, conferences. Our e-mail list.

SGGT seminar schedule. (iCal)
Date Speaker Title

Jan. 25, 1:00 p.m.

Marko Stosic (IST, Lisbon) sl(N)-link homology using foams and the Kapustin-Li formula
Feb. 1 Probably no seminar this week
Feb. 8 Jacob Rasmussen (Princeton) Lens space surgeries and L-space homology spheres
Feb. 15 Anna Beliakova (U. Zürich)

On simplification of combinatorial link Floer homology

Feb. 22, in Math 131,SUNY Stony Brook.

(Tea at 3:30 in the 4th floor common room.)

New York Area Joint Symplectic Geometry Seminar
4:00 p.m.: Junho Lee (U. Central Florida) Local Gromov--Witten invariants of Spin curve
5:15 p.m.: Octav Cornea (U. Montréal) Rigidity of monotone Lagrangians
Feb. 29 Weimin Chen (U. Massachusetts, Amherst) Group actions on homotopy K3s

March 7 at Columbia

in Math 520.

New York Area Joint Symplectic Geometry Seminar

1:00 p.m.: Dusa McDuff (Stony Brook / Columbia)

Symplectic embeddings of 4-dimensional ellipsoids and Farey numbers
2:15 p.m.: Ko Honda (U. Southern California) Contact structures, Heegaard Floer homology and triangulated categories
March 14 No seminar this week.
March 21 No seminar this week (spring break).
March 28 David Nadler (Northwestern) Springer theory for symplectic geometers
April 4 Denis Auroux (MIT) Special Lagrangian fibrations and mirror symmetry
April 11 András Juhász (Princeton) The sutured Floer homology polytope
April 18 Yaron Ostrover (MIT) On some algebraic properties of the quantum homology
April 25 Yanki Lekili (MIT) Wrinkled fibrations on near-symplectic manifolds
May 2 Maksim Lipyanskiy (MIT) A Semi-Infinite Cycle Construction of Floer Homology

# Abstracts.

#### January 25, 2008.

Marko Stosic, "sl(N)-link homology using foams and the Kapustin-Li formula" (PDF, iCal)

Abstract: In joint work with M. Mackaay and P. Vaz [4], we define an almost topological construction of a rational link homology categorifying the sl(N)-link invariant. This construction uses foams which generalize the ones introduced by Khovanov in [1]. The evaluation of closed foams uses the Kapustin-Li formula, adapted to the context of foams by Khovanov and Rozansky [2]. We conjecture that our link homology theory is equivalent to Khovanov and Rozansky's in [3].
In this talk I will present the topological aspects of this theory and show how to use the Kapustin-Li formula in order to evaluate the closed foams.

References:

[1] M. Khovanov, sl(3) link homology, Alg.Geom.Top. 4(2004), 1045-1081.
[2] M. Khovanov and L. Rozansky, Topological Landau-Ginzburg models on a world-sheet foam, hep-th/0404189.
[3] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, QA/0401268
[4] M.Mackaay, M. Stosic and P.Vaz, sl(N)-link homology using foams and the Kapustin-Li formula, arXiv:0708.2228

#### February 8, 2008.

Jacob Rasmussen, "Lens space surgeries and L-space homology spheres" (PDF, iCal)

Abstract: Which knots in S^3 have lens space surgeries? A conjecturally complete
answer to this question was given by Berge. I'll explain how Heegaard Floer
homology can be used to reformulate Berge's conjecture in terms of two other
problems: one a question about numbers, and the other a (rather different)
problem about knots in lens spaces.

#### February 15, 2008.

Anna Beliakova, "On simplification of combinatorial link Floer homology" (PDF, iCal)

Abstract: We define a new combinatorial complex computing the hat version of link Floer homology, which turns out to be simpler than Manolescu-Ozsvath-Sarkar one. We speculate that similar ideas can be used to define Seidel-Smith complex
combinatorially and show that its homology coincides with Khovanov's one.

#### February 22, 2008. (PDF)

4:00 p.m.: Junho Lee, "Local Gromov--Witten invariants of Spin curves" (PDF, iCal)

Abstract: This is a joint work with Thomas H. Parker. We define a new type of sym- plectic "local Gromov-Witten invariant" of a spin curve. When X is a Kahler surface with a smooth canonical divisor D, its (full) GW invariants are expressed in terms of such local invariants, which in turn are universal functions determined by the genera of the canonical divisor components and the holomorphic Euler characteristic of X. We also show how these local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. This yields an interesting theorem relating two- and four-dimensional Gromov-Witten theory.

5:15 p.m.: Octav Cornea, "Rigidity of monotone Lagrangians" (PDF, iCal)

Abstract: I will discuss some forms of rigidity for monotone Lagrangian submanifolds. A (very) simple example of the phenomena that I'll focus on is provided by the following statement: any two monotone Lagrangians in $CP^{n}$ intersect or one of them is small in the sense that its Gromov width is strictly smaller than that of $CP^{n}$. Joint work with Paul Biran.

#### February 29, 2008.

Weimin Chen, "Group actions on homotopy K3s" (PDF, iCal)

Abstract: K3 surface is known to have the "largest" number of exotic smooth structures (K3 with an exotic smooth structure is called a homotopy K3). In this talk we will discuss several recent results (joint with S. Kwasik) about finite group actions on homotopy K3s which reflect the following intuitive principle: exotic smooth structures are less symmetric.

#### March 7, 2008. In Math 520, Columbia. (PDF)

1:00 p.m.: Dusa McDuff, "Symplectic embeddings of 4-dimensional ellipsoids
and Farey numbers" (PDF, iCal)

Abstract: I will explain why the problem of symplectically embedding one 4-dimensional ellipsoid into another is equivalent to the problem of embedding a certain disjoint union of balls into another ball. (The basic idea is to desingularize the weighted projective plane that is obtained from the ellipsoid by collapsing its boundary along the characteristc flow.) The ball embedding problem is in principle solved; I will discuss some explicit examples. I hope also to be able to explain how one can use the same idea to construct symplectomorphisms of weighted projective planes, and hence construct some new 6-dimensional Hamiltonian S^1-manifolds.

2:15 p.m.: Ko Honda, "Contact structures, Heegaard Floer homology and triangulated categories" (PDF, iCal)

Abstract: The goal of this talk is to associate a category \mathcal{C}(\Sigma) to a surface \Sigma, called the contact category and constructed from contact structures on \Sigma x [0,1]. The category \mathcal{C}(\Sigma) satisfies many of the axioms of a triangulated category, and, in particular, has distinguished triangles which we call the {\em bypass exact triangles}. We then describe an "exact" functor from \mathcal{C}(\Sigma) to the category of vector spaces, via Heegaard/sutured Floer homology.

#### March 28, 2008.

David Nadler, "Springer theory for symplectic geometers" (PDF, iCal)

Abstract: In its modern formulation, Springer's theory of Weyl group representations involves perverse sheaves on the nilpotent cone of a Lie algebra. I will explain
a much more down-to-earth viewpoint on this subject via symplectic geometry and Floer theory. I will also discuss how it appears as a toy model of the Geometric
Langlands program.

#### April 4, 2008.

Denis Auroux, "Special Lagrangian fibrations and mirror symmetry" (PDF, iCal)

Abstract: This talk will focus on a geometric proposal for constructing the mirror of a compact Kahler manifold equipped with an anticanonical divisor, extending the Strominger-Yau-Zaslow conjecture beyond the Calabi-Yau case. The mirror manifold is constructed as a (complexified) moduli space of special Lagrangian tori, and the Landau-Ginzburg superpotential is defined by a weighted count of holomorphic discs. We will give examples, both in the toric and in the non-toric setting, to illustrate the construction and the manner in which instanton corrections arise from exceptional discs and wall-crossing phenomena.

#### April 11, 2008.

András Juhász, "The sutured Floer homology polytope" (PDF, iCal)

Abstract: Using sutured Floer homology (in short SFH) I will define a polytope inside the second relative cohomology group of a sutured manifold. This is a generalization of the dual Thurston norm polytope of a link-complement studied by Ozsvath and Szabo using link Floer homology. The polytope is maximal dimensional under certain conditions. Moreover, surface decompositions correspond to the faces of the polytope in some sense. These imply that if the rank of SFH is < 2^{k+1} then the sutured manifold has a depth at most 2k taut foliation. Moreover, SFH acts as a complexity for balanced sutured manifolds.

#### April 18, 2008.

Yaron Ostrover, "On some algebraic properties of the quantum homology"
(PDF, iCal)

Abstract: In this talk we discuss certain algebraic properties of the quantum homology algebra of toric Fano manifolds. In particular, we describe an easily-verified sufficient condition for the semi-simplicity of the quantum homology. Moreover, we provide some examples of toric Fano manifolds for which the quantum homology is not semi-simple. (This is a joint work with Ilya Tyomkin.)

#### April 25, 2008.

Yanki Lekili, "Wrinkled fibrations on near-symplectic manifolds" (PDF, iCal)

Abstract: Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. As an application, we disprove a conjecture of Gay and Kirby about essentialness of achiral singularities for broken fibrations on arbitrary closed 4-manifolds.

#### May 2, 2008.

Maksim Lipyanskiy, "A Semi-Infinite Cycle Construction of Floer Homology" (PDF, iCal)

Abstract: I will describe a new construction of Floer Theories based on a notion of "semi-infinite" geometric cycle. As opposed to the traditional Morse-theoretic approach, no transversality/nondegeneracy assumptions are necessary for the definition. I will illustrate the theory on two examples: Seiberg-Witten-Floer homology as well as Hamiltonian Floer theory for loops in C^n.

# Other relevant information.

## Our e-mail list.

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.