Workshop on Mirror Symmetry, Symplectic Geometry, and Related Topics
July 19-23, 2010, MIT

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Robert Lipshitz (Columbia):   Bordered Heegaard Floer homology

Bordered Floer homology is an extension of Ozsvath-Szabo's Heegaard Floer invariants to 3-manifolds with boundary. In the first hour we will explain the general structure of bordered Floer homology, and how it relates to the rest of the Heegaard Floer package. We will then review the definitions of HF-hat, and sketch a construction of the bordered Floer invariants. In the second hour, we will discuss some analytic aspects of the theory, and in particular sketch a proof of the main gluing property of bordered Floer homology. In the third hour, we will discuss some computations and applications.
This is joint work with P. Ozsvath and D. Thurston.

Paul Seidel (MIT):   Altering symplectic manifolds by homologous recombination

These lectures will discuss symplectic structures on noncompact manifolds, and various ways in which they can be manipulated and (using Floer cohomological or SFT invariants) distinguished. The first lecture will be an introduction to the subject, highlighting the results from Mark McLean's thesis ( The second and third lecture present more recent joint work of Abouzaid and myself.


Alexander Efimov (RAS):   HMS for P1 minus ≥3 points

Richard Garavuso (TU Wien):   Extending Hori-Vafa

Eli Grigsby (Boston College):   On Khovanov-Seidel quiver algebras, sutured Khovanov homology, and Bordered Floer homology

Ozsvath-Szabo's spectral sequence from Khovanov homology to Heegaard Floer homology has generated a number of interesting applications to questions in low-dimensional topology. By combining the constructions of Ozsvath-Szabo with sutured manifold theory, we now have an enhanced understanding of the algebraic structure of the connection. In particular, a generalization of Juhasz's surface decomposition theorem implies that the algebra of the spectral sequence behaves ``as expected" under natural geometric operations like cutting and stacking.

In this talk, I will discuss joint work in progress with Denis Auroux and Stephan Wehrli aimed at understanding how the connection between Khovanov and Heegaard-Floer homology behaves under gluing. More precisely, we will see how to recover (a portion of) the sutured version of Khovanov homology as the Hochschild homology of certain bimodules over quiver algebras defined by Khovanov-Seidel. Along the way, we will discuss an intriguing relationship between these Khovanov-Seidel bimodules and certain bimodules appearing in the bordered Floer package of Lipshitz-Ozsvath-Thurston.

Ludmil Katzarkov (U. Miami):   Noncommutative Hodge structures and applications

Yankı Lekili (MPIM Bonn):   Lagrangian correspondences and invariants for 3-manifolds with boundary I

(see the abstract for Tim Perutz)

Sikimeti Ma'u (MSRI / Columbia):   Bimodules and Lagrangian correspondences

Maksim Maydanskiy (Stanford / Cambridge):   SH = HH

Peter Ozsváth (MIT):   TBA

Tim Perutz (UT Austin):   Lagrangian correspondences and invariants for 3-manifolds with boundary II

Joint work with Yanki Lekili constructs invariants for 3-manifolds with two boundary components. We take an indefinite Morse function, locally constant on the boundary, and extract from it a sequence of Lagrangian correspondences between g-fold symmetric products of the regular levels. We prove, using algebraic geometry and cut-and-paste symplectic geometry, that the sequence is independent of the Morse function, up to equivalence in the sense of Wehrheim - Woodward. Quilted Floer cohomology techniques then supply invariants - functors between filtered Fukaya categories of compact Lagrangians - satisfying a TQFT-style gluing law. The resulting invariants for (twice punctured) closed 3-manifolds are the Heegaard Floer cohomology groups, in the "hat", "infinity", "plus" and "minus" versions. Auroux has shown that the "hat" version embeds into the bordered Heegaard theory of Lipshitz-Ozsvath-Thurston.

Jake Rasmussen (Cambridge):   Holomorphic triangles and HKM maps

Honda, Kazez, and Matic defined maps on sutured Floer homology induced by a contact 3-manifold with convex boundary. These maps were later used by Juhasz to define maps induced by four-dimensional cobordisms in sutured Floer homology. I'll explain how these HKM maps can be defined by counting holomorphic triangles, in analogy with the usual definition of cobordism maps in Heegaard Floer homology.

Helge Ruddat (Berkeley):   Mirror symmetry for non-CY hypersurfaces in toric varities

Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov we give a construction for the mirror of X for non-negative Kodaira dimension which generalizes Batyrev's mirror construction. The mirror is generally given by a reducible space equipped with a sheaf of vanishing cycles. We give evidence for the mirror duality by computing the Hodge numbers of the mirror.

Chris Woodward (Rutgers):   Gauged Floer theory of toric moment fibers

We investigate the gauged linear sigma model for holomorphic disks. We reproduce the results of Fukaya et al on non-displaceable toric moment fibers, and extend them to orbifolds.