Titles and Abstracts Mohammed Abouzaid (MIT) Functoriality in Floer theory Starting with a Landau-Ginzburg model, on a symplectic manifold, one may consider three Fukaya categories associated respectively to the fibre, the total space, and the superpotential. I will describe functors relating these three categories, and, assuming that the Landau-Ginzburg model is a Lefschetz firbation, I will derive a description of the Fukaya category of the total space, which is a priori more complicated, from the other two. As motivation, I will explain the connection with homological mirror symmetry. This is joint work with Paul Seidel. ---------------------------------------------------------------- Andrea Brini (Imperial College) Local Gromov-Witten theory and Integrable Hierarchies An influential conjecture of Witten (1990) suggests the existence of a remarkable connection between generating functions of Gromov-Witten invariants and tau functions of classical integrable hierarchies; however, concrete instances of this correspondence have proven to be hard to find and describe in detail. In this talk I will report on recent progress in the construction of new classes of examples of the Gromov-Witten/Integrable Systems correspondence which appear in the context of the Gromov-Witten theory of toric Calabi-Yau threefolds. I will also discuss implications for (and applications to) each side of the correspondence. ----------------------------------------------------------------- KwokWai Chan (HongKong) An open toric mirror theorem In this talk, I will discuss the proof of an open mirror theorem for semi-Fano toric manifolds, which states that the Floer-theoretic potential function defined by Fukaya-Oh-Ohta-Ono can be obtained from the Hori-Vafa superpotential via the mirror map. This is joint work with S.-C. Lau, N.C. Leung and H.-H. Tseng. ----------------------------------------------------------------- Alessandro Chiodo (Grenoble) LG/CY correspondence, Iritani Z-structures and Orlov equivalence In collaboration with Iritani and Ruan we prove a version of the LG/CY correspondence matching, via a symplectic transformation U_{LG/CY}, two genus-zero theories: the Fan-Jarvis-Ruan-Witten theory on the LG side and Gromov-Witten theory on the CY side. We draw inspiration from Witten's early LG/CY treatement starting from a variation of quotients in geometric invariant theory (GIT). As illustrated by Herbst, Hori and Page, this GIT transition can be used to recast Orlov's LG/CY equivalence in the phase transition between the LG orbifold point and the large volume point attached to CY hypersurfaces. In this setup, using Iritani Z-structures, we derive the symplectic transformation U_{LG/CY} from Orlov's equivalence. ----------------------------------------------------------------- Kenji Fukaya (Kyoto) Hochshild-Cyclic homology and Lagrangian Floer theory of arbitrary genus We explain how we can extend the Floer theory using the moduli space of pseudo-holomorphic map from bordered Riemann surface of arbitrary genus. It will be certain algebraic structure called involutive bi Lie infinity structure on the cyclic bar complex of de Rham complex. I will explain how it is used to show Hodge to de Rham degenration of Cyclic homology of Fukaya category in certain sense and how to use it to study Gromov-Witten invariant of the symplectic manifold using Floer theory of its Lagrangian submanifold. ----------------------------------------------------------------- Hiroshi Iritani (Kyoto) Gamma structure and functoriality in Gromov-Witten theory I will discuss the Gamma structure in Gromov-Witten theory and its relationships to the functoriality problem. ----------------------------------------------------------------- Tyler Jarvis (BYU) Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras I will discuss recent work with Drew Johnson, Amanda Francis, and Rachel Suggs on the Landau-Ginzburg Mirror Symmetry Conjecture for orbifolded Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory (Landau-Ginzburg A-model) of the orbifold [W/G] is isomorphic as a Frobenius algebra to the orbifold Milnor ring of [W^T/G^T] (as defined by Kaufmann and Krawitz), associated to the dual polynomial W^T and dual group G^T. ----------------------------------------------------------------- Jun Li (Stanford) Cosection localized virtual cycles and applications to LG theory We show that the cosection localized virtual cycle is the natural algebraic version of virtual cycle using perturbed equation on non-compact moduli spaces. Using this, we have (re)constructed the LG theory of the K_{P4}, of quasi-homogeneous polynomials (earlier by Fan-Javis_Ruan). This is a combination of joint work with HL Chang, YH Kiem and WP Li. ----------------------------------------------------------------- Si Li (Northwestern) Quantum A_1 singularity and pure gravity The quantization of BCOV theory leads to higher genus B-model on Calabi-Yau manifolds. I'll describe a parallel construction with the existence of superpotential. For the simplest example of A_1 singularity, we show that there's a unique quantization compatible with Virasoro equations. In particular, the partition function will recover the Witten-Kontsevich tau-function for pure gravity. ------------------------------------------------------------------ Yongbin Ruan (Michigan) Gromov Witten theory of quotient of quintic 3-fold via global mirror symmetry Suppose that $X$ is a quintic 3-fold defined by a quintic polynomial $W$. There are two outstanding conjectures: (i) Landau-Ginzburg/Calabi-Yau correspondence governing the equivalence of Gromov-Witten theory of $X$ and the so called FJRW-theory of $(W, Z_5)$. (ii) The modularity of Gromov-Witten theory of $X$. Both problems are expected to be true for the finite quotient of $X$. In the talk, we describe a on-going program to solve the above conjectures for $X/Z^5_5$. We do so by the method of "global mirror symmetry" which consists of a rigorous construction of higher genus B-model theory of its Landau-Ginzburg mirror and two mirror theorems of all genera. This is a joint work with Iritani, Milanov and Shen. ------------------------------------------------------------------ Sergey Shadrin (UVA, Amstedam) Cohomological field theories and integrable hierarchies. I'll make a survey of several different ways to associate an integrable hierarchy to a semi-simple cohomological field theory and will discuss what is known about the correspondences between different constructions. ------------------------------------------------------------------ Nick Sheridan (MIT) Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. After stating the result carefully, we will describe some of the techniques used in its proof, and draw lots of pictures in the one-dimensional case. ----------------------------------------------------------------- Atsushi Takahashi (Osaka) Classical mirror symmetry between orbifold projective lines and cusp singularities We report on our recent study on the classical mirror symmetry between orbifold projective lines and cusp singularities, an isomorphism of Frobenius manifolds from the Gromov--Witten theory for an orbifold projective line and the one via a primitive form associated to the unfolding of the cusp singularity.