Tobias Ekholm (Uppsala):
Contact surgery on a contact manifold Y along a Legendrian sphere Λ in Y gives a new contact manifold Y_{Λ} with a new Legendrian sphere Γ in Y_{Λ} (the co-core sphere). We find an exact triangle relating the linearized contact homology of Y to that of Y_{Λ} via a cyclic version of the Legendrian contact homology of Λ. Furthermore, we compute the Legendrian contact homology of Γ in terms of that of Λ.
In my third talk I will discuss joint work with Etnyre and Sabloff on the following subject.
A long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself is established. In this sequence, the singular homology H_{*} maps to linearized contact cohomology CH^{*} which maps to linearized contact homology CH_{*} which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_{*} -> H_{*}) and the cokernel of the map (H_{*} -> CH^{*}). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_{*} maps to a class α in CH^{*} such that α(a)=1.
Katrin Wehrheim (MIT), Chris Woodward (Rutgers), Sikimeti Ma'u (Rutgers): Holomorphic Quilts: Functoriality for Lagrangian correspondences in Floer-Fukaya theory
We describe a calculus of functors for Lagrangian correspondences in Floer-Fukaya theory, which plays the role of ``mirror dual'' of the calculus of Mukai functors for derived categories of coherent sheaves. The calculus is used to partially construct a category-valued field theory for three-dimensional cobordisms possibly containing tangles.Lecture 1 (Wehrheim): Lagrangian Correspondences and Holomorphic Quilts
A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was introduced by Weinstein in an attempt to build a symplectic category that has morphisms between any pair of symplectic manifolds (not just symplectomorphic pairs). We define such a category, in which all (generalized) Lagrangian correspondences are composable morphisms. We extend it to a 2-category by extending Floer homology to generalized Lagrangian correspondences. This is based on counts of 'holomorphic quilts' -- a collection of holomorphic curves in different manifolds with 'seam values' in the Lagrangian correspondences. A fundamental isomorphism of Floer homologies ensures that our constructions are compatible with the geometric composition of Lagrangian correspondences.Lecture 2 (Wehrheim): Floer Field Theory
We give a general prescription for constructing topological invariants and TQFT's by "decomposition into simple pieces" and show examples in 2+1 dimension, using gauge theoretic representation spaces. We consider e.g. 3-manifolds or links as morphisms (cobordisms or tangles) in a topological category. In order to obtain a topological invariant from our generalized Floer homology, it suffices to (i) decompose morphisms into simple morphisms (e.g. by cutting between critical levels of a Morse function) (ii) associate to the objects and simple morphisms smooth symplectic manifolds and Lagrangian correspondences between them (e.g. using moduli spaces of bundles or representations) (iii) check that the moves between different decompositions are associated to (good) geometric composition of Lagrangian correspondencesLecture 3 (Ma'u): Gluing quilted disks
I'll describe how to get an A_{infty} functor between the Fukaya categories of two symplectic manifolds out of a Lagrangian correspondence between the manifolds. The functor is constructed using ``quilted disks'' with the given Lagrangian correspondence as the ``seam condition''. The proof of the functor axiom depends on a gluing statement that relates broken (isolated) quilted disks with ends of the one-dimensional component, which generalizes the gluing result needed for the construction of the Fukaya category in the exact and monotone cases.Lecture 4 (Woodward): Seidel's exact triangle and its generalization to fibered twists.
Seidel's exact triangle (conjectured by Kontsevich) is a ``categorification'' of the Picard-Lefschetz formula for the monodromy of a Lefschetz fibration. I will (i) describe Seidel's result and (ii) describe a generalization to case that the fibration has Morse-Bott singularities, which appears naturally in various situations such as nilpotent slices and moduli spaces of flat bundles. In the case of SU(n) Lagrangian Floer theory for tangles, one obtains an exact triangle for the tangle invariants which looks like that of Khovanov-Rozansky. A similar result has been developed by T. Perutz, for the case of symmetric products of curves.References:
On the level of Floer homology the calculus of functors is described in: Katrin Wehrheim and Chris T. Woodward. Functoriality for Lagrangian correspondences in Floer theory. arXiv:0708.2851.Mohammed Abouzaid (Clay Institute and MIT): Wrapped Fukaya categories and string topology
I'll discuss how the most elementary form of string topology (Pontrjagin product on the loop space) gives a model not just for the wrapped Floer homology of a cotangent fibre, but for the entire wrapped Fukaya category of cotangent bundles. This gives an elementary way of defining a restriction functor from the Fukaya category of a symplectic manifold to the wrapped category of an exact Lagrangian. I will likely point out some difficulties in adapting this to more complicated symplectic manifolds, and some strategies for resolving them.Ezra Getzler (Northwestern): Open-closed topological field theories
A close examination of Harer's triangulation of Teichmuller space establishes a filtration F_{k}, k=0,1,...,2g-2+n, of the Harvey bordification, and hence of the modular operad governing topological field theories in two dimensions, with the following property: the pair (F_{k},F_{k-1}) is k-connected. This generalizes a series of theorems, including Turaev's characterization of G-equivariant topological fields theories in two dimensions (this follows from the case k=1), the Moore-Seiberg theorem (k=2), and open-closed generalizations of these results, due to Moore and Segal and others.Eleny Ionel (Stanford): Symplectic degenerations and Gromov-Witten invariants
Mikhail Khovanov (Columbia): How to categorify a quantum group
I'll explain a recent joint work with Aaron Lauda on categorification of positive halves of quantum Kac-Moody algebras.Maksim Lipyanskiy (MIT): A semi-infinite cycle construction of Floer theory
We present a new foundation for Floer theory closer in spirit to bordism than Morse theory. After discussing the general framework, we will illustrate the theory on examples from symplectic geometry and gauge theory as well as discuss the relationship to the traditional Morse-Floer approach.Jacob Lurie (MIT): Topological field theories and string topology
I will describe joint work with Mike Hopkins which provides a classification of 2-dimensional topological quantum field theories, building on earlier work of Maxim Kontsevich and Kevin Costello. A special case of this result allows us to recover the Chas-Sullivan "string topology" operations on the homology of the loop space of a compact manifold.Ciprian Manolescu (Columbia): Symplectic instanton homology
Starting from a Heegaard decomposition of a three-manifold, we use Lagrangian Floer homology to construct a symplectic analog of the tilde version of instanton homology. This is joint work with Chris Woodward.Mark McLean (Cambridge): Exotic Stein manifolds
A Stein manifold is a complex manifold with a closed holomorphic embedding in C^{n}. This has a symplectic form induced from the standard one in C^{n}. In each complex dimension greater than three, I will construct infinitely many Stein manifolds diffeomorphic to Euclidean space which are pairwise distinct as symplectic manifolds. I will distinguish them using an invariant called symplectic homology.Tomasz Mrowka (MIT): Knot invariants from instantons
Rahul Pandharipande (Princeton): Descendent integrals for open Riemann surfaces
I will talk about work with J. Solomon related to integration of cotangent line classes over moduli spaces of open Riemann surfaces. In genus 0, there is a complete theory defined geometrically, and we have solved it. In higher genus, the foundations are still being sorted out, but nevertheless we've solved the theory. The answer is a remarkable set of equations which contain Witten's KdV for closed integrals.Atsushi Takahashi (Osaka): HMS for isolated hypersurface singularities
Homological mirror symmetry conjecture for graded isolated hypersurface singularities will be explained. After reviewing some necessary basic definitions such as Fukaya categories and categories of graded matrix factorizations, I'll formulate HMS conjecture and then give examples and relevant results. Relation between weighted projective lines and cusp singularities will also be discussed.Valerio Toledano Laredo (Northeastern): Stability conditions and Stokes factors
D. Joyce recently defined invariants counting semistable objects in an abelian category A with a given class in K(A). He obtained wall-crossing formulae with respect to a change of stability condition for these invariants, constructed holomorphic generating functions for these and showed that they satisfy an intriguing non-linear PDE.I will explain how Joyce's wall-crossing formulae may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of the category A. This allows one in particular to interpret his generating functions as defining an isomonodromic family of such connections parametrised by the space of stability conditions of A.
This is joint work with T. Bridgeland (arXiv:0801.3974).
Kazushi Ueda (Osaka): Toric degenerations of Gelfand-Cetlin systems and potential functions
(joint work with Takeo Nishinou and Yuichi Nohara)Gelfand-Cetlin systems are completely integrable Hamiltonian systems on generalized flag manifolds of type A. Their momentum polytopes are called Gelfand-Cetlin polytopes, whose integral points are in bijection with Gelfand-Cetlin bases in representation theory.