Abstracts of Talks

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Homological Mirror Symmetry for T⁴

Mohammed Abouzaid

I shall explain how the pseudo-holomorphic quilt techniques of Wehrheim and Woodward yield of proof of Homological mirror symmetry for products, assuming mirror symmetry is known for both factors, and stringent conditions are imposed on the behaviour of holomorphic curves. An application to Lagrangian embeddings in T⁴ will be presented. This work was done jointly with Ivan Smith.

Lagrangian Fibrations and Mirror Symmetry for Blowups

Denis Auroux

This talk is a report on joint work in progress with Mohammed Abouzaid and Ludmil Katzarkov, on the construction of Landau-Ginzburg mirrors for blow-ups from the perspective of the Strominger-Yau-Zaslow conjecture. We will first recall how, in the SYZ framework, the mirror is constructed from a (special) Lagrangian torus fibration and the superpotential arises as a Floer-theoretic obstruction. We will then consider specifically the case of the blowup of a toric variety along a codimension 2 subvariety contained in a toric hypersurface. We will discuss the SYZ mirror and its instanton corrections, as well as the superpotential. This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, curves of arbitrary genus, etc.

Landau-Ginzburg/Calabi-Yau Correspondence for Quintic three-folds via Symplectic Transformations

Alessandro Chiodo

This talk is based on work in collaboration with Yongbin Ruan. We compute the Fan-Jarvis-Ruan-Witten theory in genus zero for quintic polynomials in five variables and show that it is encoded into an I-function satisfying the same Picard-Fuchs equation as that of the quintic three-fold. This implies that the two I-functions define two bases of the solution space of the Picard-Fuchs equation. Then, we establish a form of Landau-Ginzburg/Calabi-Yau correspondence by computing explicitly the change of basis matrix (symplectic transformation).

A Remark on a Conjecture of Hain and Looijenga

Carel Faber

After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, I will prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true.

Mirror Symmetry for P² and Tropical Geometry

Mark Gross

I will explain how tropical geometry can be used to describe the Landau-Ginzburg mirror for P² in a natural way using tropical geometry. In particular, flat coordinates on the universal unfolding moduli space for the Landau-Ginzburg potential can be described naturally. In addition, this approaches gives a simple explanation of why this mirror symmetry works.

Constructing the Virtual Fundamental Cycle in the Landau-Ginzburg A model

Huijun Fan

In this talk, I will describe how to use the analytical technique to construct the virtual fundamental cycle in the module space of W-curves via the Witten equation, where W is a quasi-homogeneous polynomial. These techniques including the nonlinear estimate of P.D.E., Fredholm index theory, Gluing technique and the others. This is the main analytical part of the joint work with Yongbin Ruan and Tyler Jarvis.

The Witten Equation, Mirror Symmetry and Quantum Singularity Theory

Tyler Jarvis

I will describe recent joint work with Huijun Fan and Yongbin Ruan in which we construct, for every non-degenerate quasi-homogeneous singularity, a moduli space of decorated stable curves and a virtual cycle on that space. This provides an "orbifold Landau-Ginzburg A-model" and a corresponding cohomological field theory for each of these singularities. In the special case of the A_{r-1} singularity, our constructions match the theory of r-spin curves.

For many simple singularities the resulting Frobenius algebra is mirror dual to the Milnor ring of the singularity. For other singularities, a more complicated mirror symmetry relation has been conjectured by Krawitz. This conjecture has been verified in many cases.

Finally, I will describe our proof of the Witten ADE-integrable hierarchy conjecture, which states that the potential functions for our theory for simple singularities (A,D, and E) satisfy corresponding integral hierarchies.

Homological Mirror Symmetry and Birational Geometry

Ludmil Katzarkov

In this talk we will discuss a possible new approach to questions of nonrationality and rigidity in Birational geometry.

Orbifold Landau-Ginzburg theories, Mirror symmetry and Frobenius Structures

Ralph Kaufmann

We present old and new aspects of orbifold Landau-Ginzburg theories treated as singularities with symmetries.

Orbicurves in Gromov-Witten Theory and Higher Spin Theory

Takashi Kimura

We explain the analogy between higher spin theory (in the sense of Jarvis-Kimura-Vaintrob) and orbifold Gromov-Witten theory (due to Chen-Ruan and Abramovich-Graber-Vistoli), two different constructions of cohomological field theories. Higher spin theory has subsequently been generalized to a kind of Landau-Ginzburg A-model by Fan-Jarvis-Ruan. This analogy suggests some interesting new directions in these theories.

W-constraints for Singularities of Type A_N

Todor Milanov

In this talk I am planning to describe an approach whose goal is to characterize Gromov--Witten invariants via differential operators constraints similar to the Virasoro constraints. The idea will be explained in the case of A_N-singularity. Our methods are based entirely on the symplectic loop space formalism of A. Givental and thus I am expecting that they can be applied successfully to the mirror models of symplectic manifolds.

Homological Mirror Symmetry for the Genus Two Curve

Paul Seidel

I will survey what we know about homological mirror symmetry in the lowest-dimensional case (corresponding to the symplectic geometry of surfaces), including some recent developments.

We explain the analogy between higher spin theory (in the sense of Jarvis-Kimura-Vaintrob) and orbifold Gromov-Witten theory (due to Chen-Ruan and Abramovich-Graber-Vistoli), two different constructions of cohomological field theoriesË. Higher spin theory has subsequently been generalized to a kind of Landau-Ginzburg A-model by Fan-Jarvis-Ruan. This analogy suggests some interesting new directions in these theories.

A-twisted Landau-Ginzburg Models, Gerbes, and Kuznetsov's Homological Projective Duality

Eric Shapre

In this talk we will summarize several recent developments related to Landau-Ginzburg models. We will begin by describing how one A-twists a Landau-Ginzburg model in physics, and how a physical process known as "renormalization group flow" sometimes identifies correlation functions in such A-twisted Landau-Ginzburg models with ordinary Gromov-Witten invariants. Then, after briefly reviewing how one associates a CFT to a gerbe and associated technical issues, we will outline some applications of CFT's of gerbes, including Landau-Ginzburg (Toda) mirrors to gerbes on projective spaces, and a physical realization of Kuznetsov's homological projective duality, which provides examples of LG's and GLSM's describing non-birational spaces on the same GLSM Kahler moduli space, as well as examples in which the geometry arises via nonperturbative effects, rather than the usual complete intersection story.

Commutativity Equations

Dimitri Zvonkine

A solution of the commutativity equations is described by a vector bundle with a pencil of flat connections; this is analogous to the way in which a solution of the associativity equations is described by a Frobenius manifold. We show how solutions of the commutativity equations arise in algebraic geometry; describe their relation with the Losev-Manin compactification of the moduli space M_{0,n}, and describe an analog of Givental's group action on such solutions. We also explain how this formalism allows one to predict the existence of certain universal expressions in the Gromov-Witten theory.

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