Workshop on Mirror Symmetry, Symplectic Geometry, and Related Topics
June 22-26, 2009, MIT

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Peter Kronheimer (Harvard):   Instantons, Floer homology and applications

Donaldson's polynomial invariants of smooth 4-manifolds were introduced in 1986, and the accompanying invariants of 3-manifolds -- Floer's instanton homology -- followed soon after. In these three lectures, I will review the outlines of Floer's construction, and give an updated account of how instanton homology can be used. In particular, there are many more recent developments (in Heegaard Floer theory and Seiberg-Witten theory) which motivate new constructions on the instanton side. For example, one can construct an instanton knot homology which is a categorification of the Alexander polynomial. Instanton homology can also be used to detect fibered 3-manifolds (following Ni), detect the Thurston norm (following ideas from Juhasz taken from the Heegaard Floer setting), and prove that non-trivial knots have Property P. An improved understanding of instanton homology also sheds light on the polynomial invariants of 4-manifolds, providing alternative and more powerful techniques for calculation.

Tom Coates (Imperial College London) and Hiroshi Iritani (Kyushu U.):   Wall-crossings in toric Gromov-Witten theory

It has been conjectured by Yongbin Ruan that two spaces that are related by a crepant birational transformation (such as a flop, a crepant resolution, or a K-equivalence) have equivalent Gromov-Witten theories. We will explain what "equivalent" means, and how mirror symmetry for toric orbifolds implies the conjecture at genus zero. We will then outline how the relationship between the two Gromov-Witten theories here arises from a Fourier-Mukai transformation. If time permits, we will discuss the higher-genus generalization of Ruan's conjecture.

Large parts of this are based on joint work with Alessio Corti and Hsian-Hua Tseng.


Mohammed Abouzaid (Clay Institute and MIT):  A chain level model for the Fukaya category of a plumbing

I will explain how an A infinity equivalence can be setup between the Floer cochains of an exact Lagrangian and its singular cochains in such a way that it can be extended to Lagrangian skeleta consisting of smooth Lagrangians meeting (sufficiently) nicely.

Kwok Wai Chan (Harvard):  Strominger-Yau-Zaslow transformations in mirror symmetry

I will discuss how SYZ transformations were applied to the study of mirror symmetry for semi-flat Calabi-Yau manifolds and toric Fano manifolds.

Amin Gholampour (Caltech):  BPS invariants for resolutions of polyhedral singularities

Let G be a finite subgroup of SO(3). Y=G-Hilb(C3) gives the preferred Calabi-Yau resolution of singularities of C3/G. We use two different approaches to determine the BPS invariants of Y: Gromov-Witten theory, and a specific moduli space of torsion sheaves on Y. The result is expressed in terms of an ADE root system canonically associated to G.

Tamas Hausel (Oxford):  Mirror symmetry, Langlands duality and the Hitchin system

I will survey developments concerning topological mirror symmetry for Hitchin systems for SLn and its Langlands dual PGLn. I will concentrate on recent work with de Cataldo and Migliorini on relating the topological mirror symmetry conjectures for Hitchin systems and character varieties and ultimately to Ngo's work on the fundamental Langlands Lemma.

Ludmil Katzarkov (U. of Miami):  Homological mirror symmetry and DG-schemes

Bumsig Kim (KIAS):  Some compactifications of moduli of maps from curves

Some methods to compactify moduli of maps from curves will be presented.

Chiu-Chu Melissa Liu (Columbia):  The coherent-constructible correspondence and toric homological mirror symmetry

I will discuss (i) SYZ transformation relating equivariant coherent sheaves on the toric variety to Lagrangians in the cotangent of Rn, (ii) microlocalization functor relating the Fukaya category of the cotangent to constructible sheaves on the base (due to Nadler-Zaslow, Nadler), and (iii) a categorification of Morelli's theorem relating equivariant coherent sheaves on the toric variety to constructible sheaves on Rn. This is a joint work with Bohan Fang, David Treumann, and Eric Zaslow.

Lenny Ng (Duke):  Algebraic aspects of Legendrian Symplectic Field Theory

I will discuss the algebraic structure of the recently-developed invariant of Legendrian knots in R3 obtained by counting boundary-punctured holomorphic disks. This involves curved dg-algebras and a related complex that may have something to do with sutured contact homology.

Paolo Rossi (Ecole Polytechnique):  Gromov Witten theory of orbicurves and their mirror model

As known already from the first computation of GW invariants of target smooth curves by Okounkov and Pandharipande, the topological field theory approach to GW theory for target dimension 1 (via pair of pants decomposition) is very effective. Symplectic Field Theory adds the computational power of integrable systems to this approach, especially in genus 0. Once a mirror model is identified, then, a Lax representation for the integrable system associated with the Frobenius manifold becomes available and looking for the right quantization of the Lax pair leads to higher genus expansion.

Josh Sabloff (Haverford):  A-infinity structures for Legendrian contact homology

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, I will introduce invariant cup and Massey products * and, more generally, an A-infinity structure * on the linearized LCH. I will apply the products and A-infinity structure to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors (and maybe a few more applications). This is joint work with G. Civan, J. Etnyre, P. Koprowski, and A. Walker.

Dylan Thurston (Columbia):  Bimodules and mapping class group actions in Heegaard Floer homology

Hsian-Hua Tseng (Wisconsin):  On the decomposition conjecture of étale gerbes

Let G be a finite group. A G-gerbe over a space X may be intuitively thought of as a fiber bundle over X with fibers being the classifying space (stack) BG. In particular BG itself is the G-gerbe over a point. A more interesting class of examples consist of G-gerbes over BQ, which are equivalent to extensions of the finite group Q by G. Considerations from physics have led to conjectures asserting that the geometry of a G-gerbe Y over X is equivalent to certain "twisted" geometry of a "dual" space Y'. A lot of progresses have be made recently towards proving these conjectures in general. In this talk we'll try to explain theses conjectures in the elementary concrete examples of G-gerbes over a point or BQ.

Ben Webster (MIT):  Representation theory and a strange duality for symplectic varieties

In recent work with Braden, Licata and Proudfoot, we showed that certain algebras constructed from hyperplane arrangements have a number of nice properties which are surprisingly reminiscent of the BGG category O; in particular, they are Koszul, and Koszul duality corresponds to a well known combinatorial duality. I'll explain why we think properties are connected to a geometric origin for both these categories, and how this suggests an underlying duality between pairs of symplectic varieties.

Katrin Wehrheim (MIT):  Calculations of Floer homology by reduction

We have some examples of calculating monotone Floer homology from a general strip shrinking isomorphism in quilted Floer homology (for sequences of Lagrangian correspondences). Examples include the Clifford torus in CPn (previously known by Cho) and nondisplaceable Tn-k×S2k-1 in CPn×CPk-1. Moreover, the bijection of trajectory moduli spaces can be somewhat generalized to multiply covered compositions of correspondences, yielding e.g. calculations of the Floer homology between Clifford tori and RPn in CPn (confirming work by Allston). Finally, "figure eight" bubbling obstructions can be understood explicitly. Work is in progress on overcoming these for the Chekanov/Polterovich torus in S2×S2; using symmetries and twisted coefficients.