Mohammed Abouzaid

An open string analogue of Viterbo functoriality

In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called “wrapped Floer cohomology”. We construct an A1 ‐structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A1 ‐ homomorphism realizing the restriction to a Liouville subdomain. The construction of the A1 ‐structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations. 53D40

Homological mirror symmetry for punctured spheres

We prove that the wrapped Fukaya category of a punctured sphere (S with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

Homological mirror symmetry for the 4-torus

We use the quilt formalism of Mau-Wehrheim-Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard 4-torus. As an application, we study Lagrangian genus two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of such a surface with linear Lagrangian 2-tori in in the 4-torus.

Parametrized ring-spectra and the nearby Lagrangian conjecture

Let L be an embedded closed connected exact Lagrangian submanifold in a connected cotangent bundle T N . In this paper we prove that such an embedding is, up to a finite covering space lift of T N , a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N . The homology of FL will be the (twisted) symplectic cohomology of T L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL. The fiber-wise ring structure combined with the intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L! N is always a homotopy equivalence. 53D12; 55R70, 55T10

The symplectic arc algebra is formal

This is the author accepted manuscript. The final version is available from Duke University Press via http://dx.doi.org/10.1215/00127094-3449459

On the immersion classes of nearby Lagrangians

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.