Next semester's web page.

The SGGT seminar meets on Fridays in Math 520, at 1:10 p.m, except as noted. There is also an informal symplectic geometry seminar, which meets at a different time.

Previous semesters: fall 2008, spring 2008, fall 2007, spring 2007, fall 2006.

Other area seminars. Our e-mail list.

SGGT seminar schedule. (iCal)
Date Speaker Title
Jan. 23, 1:00 p.m.

Organizational meeting, probably.
Jan. 30 Yanki Lekili (MIT) Heegaard Floer homology of broken fibrations
Feb. 6 John Etnyre (Georgia Tech) Duality exact sequences in contact homology
Feb. 13 Sucharit Sarkar (Princeton) Constructing CW complexes from grid diagrams
Feb. 20

Mikhail Gromov (Courant / IHÉS)

(Joint with GT seminar.)

 Homological Variations on Measure and Probability

Feb. 27

3:45 p.m.

 John Baldwin (Princeton) Capping off open books and the Ozsvath-Szabo contact invariant
March 6 No seminar this week.
March 13 Alexander Ritter (MIT) Novikov-symplectic cohomology and exact Lagrangian embeddings
March 27 Yakov Savelyev (U. Massachussetts, Amherst) Flow categories and Dirac loopspace
April 3 Zoltán Szabó (Princeton) Heegaard Floer homology and complete decomposing systems
April 10 No seminar this week.
April 17 1:10 p.m.: Joeseph Johns (NYU) Lefchetz fibrations on cotangent bundles and Lagrangian submanifolds
3:45 p.m.: Daniel Mathews (Stanford) Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori
April 24 Vladimir Chernov (Dartmouth) Legendrian links, causality, and the Low conjecture
May 1 NY Joint Symplectic Geometry Seminar
  
1:10 p.m.: Eric Zaslow (Northwestern) T-Duality and the Coherent-Constructible Correspondence
3:45 p.m.: Claude LeBrun (SUNY Stony Brook) Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds
May 8, in Math 507 Maksim Maydanskiy (MIT) "Exotic" symplectic manifolds via Lefschetz fibrations
May 15, in Math 507 Mikio Furuta (U. Tokyo) Polarizations and the moduli space of flat connections on a Riemann surface

 

Abstracts.

January 30, 2009.

Yanki Lekili, "Heegaard Floer homology of broken fibrations" (PDF, iCal)

Abstract: We will outline a programme for identifying Perutz's Lagrangian matching invariants and Ozsvath-Szabo's Heegaard Floer invariants of three and four manifolds. In this talk, we will deal with purely Heegaard Floer theoretical side of this programme and describe the isomorphism of 3-manifold invariants for certain Spin^c structures where the groups involved can be formulated in the language of Heegaard Floer theory. As applications, we give new calculations of Heegaard Floer homology of certain classes of 3-manifolds and an outline of a proof of Floer's excision theorem in the context of Heegaard Floer homology.

 

February 6, 2009.

John Etnyre, "Duality exact sequences in contact homology" (PDF, iCal)

Abstract: I will discuss a "duality" among the linearized contact homology groups of a Legendrian submanifold in certain contact manifolds (in particular in Euclidean (2n+1)-space). This duality is expressed in a long exact sequence relating the linearized contact homology, linearized contact cohomology and the ordinary homology of the Legendrian submanifold. One can use this structure to ease difficult computations of linearized contact homology in high dimensions and further illuminate the proof of cases of the Arnold Conjecture for the double points of an exact Lagrangian in complex n-space.

 

February 13, 2009.

Sucharit Sarkar, "Constructing CW complexes from grid diagrams" (PDF, iCal)

Abstract: Given a knot presented in a grid diagram, we construct a CW complex which has one cell for each generator of the grid chain complex and whose attaching maps correspond to the chain complex boundary map. Thus the homology of this space is the knot Floer homology. We will show that the stable homotopy type of this CW complex is actually a knot invariant.

 

February 27, 2009, 3:45 p.m.

John Baldwin, "Capping off open books and the Ozsvath-Szabo contact invariant" (PDF, iCal)

Abstract: If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. I'll describe a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and I'll discuss various applications.

 

March 13, 2009.

Alexander Ritter, "Novikov-symplectic cohomology and exact Lagrangian embeddings" (PDF, iCal)

Abstract: We are interested in finding topological obstructions to the existence of
exact Lagrangian submanifolds L inside a cotangent bundle T^*N. Under mild
homotopy assumptions on N, I proved that the image of \pi_2(L) inside \pi_2(N)
has finite index. This result makes no assumption about the Maslov class of L,
and the manifolds need not be orientable. My approach builds on Viterbo's work:
by using symplectic cohomology we construct a transfer map on the Novikov
homologies of the free loop spaces of N and L. The result then follows from a
vanishing result for the Novikov homology of loop spaces.

 

April 17, 2009.

1:10 p.m.: Joeseph Johns, "Lefchetz fibrations on cotangent bundles and Lagrangian submanifolds" (PDF, iCal)

Abstract: Given a Morse function f: N --> R I will describe a Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle D(T*N). I will then describe a program in progress for studying closed exact Lagrangian submanifolds of T*N using pi. The idea is to translate questions about Lagrangian submanifolds into questions about representations of certain quivers, following Seidel's work on T*S^n.

If time permits I will discuss the following offshoots of this program: 1. The program yields a conjectural bridge between The analysis of Fuk(T*N) by Nadler-Zaslow, using constructible sheaves, and that of Seidel, using Picard-Lefschetz theory. 2. In a more geometric vein the construction of pi: E ---> C suggests a way to generalize matching paths from spheres to more general manifolds.

3:45 p.m.: Daniel Mathews, "Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori" (PDF, iCal)

Abstract: I will talk about recent investigations of contact elements in the sutured Floer homology of solid tori, as part of the (1+1)-dimensional TQFT defined by Honda-Kazez-Matic. The Z_2 sutured Floer homology vector spaces in this case form a "categorification of Pascal's triangle", a triangle of vector spaces, with contact elements corresponding to chord diagrams and forming distinguished subsets of order given by the Narayana numbers. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams --- so that this talk is actually very elementary.

I will show that there are natural "creation and annihilation operators" which allow us to define a QFT-type basis consisting of contact elements; and contact elements are in bijective correspondence with comparable pairs of basis elements with respect to a certain partial order, and in a natural and explicit way. I will explain how we can use this to extend Honda's notion of contact category to a 2-category, and possibly a 3-category.

 

April 24, 2009.

Vladimir Chernov, "Legendrian links, causality, and the Low conjecture" (PDF, iCal)

Abstract: Let (X^{m+1}, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of R^m. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events x,y\in X are causally related if and only if the Legendrian link of spheres S_x, S_y whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m=2 the events x,y are causally related if and only if S_x, S_y is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X, g) such that the universal cover of its Cauchy surface is diffeomorphic to an open domain of R^m. An interesting fact, proved in the joint work with Yuli Rudyak, is that a certain weakened version of the Low conjecture is true for all nonrefocussing globally hyperbolic spacetimes. This includes all the cases where a Cauchy surface has infinite fundamental group or is not a closed manifold. (This is based on the joint work with Stefan Nemirovski.)

 

May 1, 2009. NY Area Joint Symplectic Geometry Seminar (PDF)

1:10 p.m.: Eric Zaslow, "T-Duality and the Coherent-Constructible Correspondence" (PDF, iCal)

Abstract: I will describe a triangle of equivalences between three categories: a) coherent sheaves on a toric variety; b) a Fukaya category of Lagrangian submanifolds of a symplectic vector space; and c) a category of constructible sheaves on a real vector space. The categories (a), (b), and (c) are the vertices of the triangle. The edge (ab) can be thought of as T-duality; (bc) can be thought of as microlocalization; and (ac) is the coherent-constructible correspondence. The edges are based on work of many authors, including (but not limited to) joint work with Nadler (bc) and Fang, Liu and Treumann (ac).

To be concrete, the edge (ac) is based on the familiar assignment of a polytope to an ample line bundle on a toric variety -- e.g., the hyperplane bundle on projective n-space gives a primitive simplex in n dimensions.

(2:15 p.m.: Sabin Cautis in the Algebraic Geometry Seminar.)

3:45 p.m.: Claude LeBrun, "Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds" (PDF, iCal)

Abstract: An Einstein metric is by definition a Riemannian metric of constant Ricci curvature. One would like to completely determine which smooth compact n-manifolds admit such metrics. In this talk, I will describe recent progress regarding the 4-dimensional case. These results specifically concern 4-manifolds that also happen to carry either a complex structure or a symplectic structure.

 

May 8, 2009, in Math 507.

Maksim Maydanskiy "'Exotic' symplectic manifolds via Lefschetz fibrations" (PDF, iCal)

Abstract: Stein manifolds are known to symplectic geometers as Liouville domains
and are an especially nice class of open symplectic manifolds. I construct, in all odd complex dimensions, pairs of Lioville domains W_0 and W_1 which are diffeomorphic to the sphere cotangent bundle with one extra subcritical handle, but are not exact symplectomorphic. In fact, while W_0 is symplectically very similar to the cotangent bundle itself, W_1 is more unusual, and in particular contains no compact exact Lagrangian submanifolds. Constructions are given by explicit Lefschetz fibrations, and the proofs involve calculations of wrapped Floer homologies.

 

May 15, 2009, in Math 507.

Mikio Furuta, "Polarizations and the moduli space of flat connections on a Riemann surface" (PDF, iCal)

Abstract: The Verlinde formula computes the dimensions of conformal blocks which are given by the quantization in a Kahler polarization of the moduli space of flat connections on a Riemann surface. In the early 90's Jeffrey and Weitsman showed that the Verlinde formula for the SU(2)-WZW model matched the quantization in a real polarization of the moduli space associated to a pants decomposition. In this talk I will explain the matching directly for the Riemann surface of genus 2 with a marked point. Our approach relies on a version of Witten deformation. Joint work with Takahiko Yoshida and Hajime Fujita.

 

Other relevant information.

Other area seminars.

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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.