Speaker: Alexander Polishchuk (University of Oregon)
Title: Matrix factorizations and Cohomological Field Theories
Abstract: We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.

Speaker: Jérémy Guéré (Humboldt University Berlin)
Title: I) Virtual classes from matrix factorizations
        II) A Landau-Ginzburg mirror theorem without concavity
Abstract: This series of lectures deals with a quantum theory for polynomial singularities. It was first described by Fan, Jarvis, and Ruan, after foundational ideas of Witten. Then, Polishchuk and Vaintrob proposed an algebraic version of this theory using matrix factorizations.

In the first lecture, I start with a cohomological description of the singularity and I state an important relation known as cohomological mirror symmetry. Then, I focus on the quantum theory of the singularity x^r, also called r-spin theory, and define the notion of quantum product to generalize mirror symmetry to an isomorphism of rings.

In the second lecture, I define the notion of an orbifold curve and use it for the moduli space of the quantum singularity theory. I treat in details the D_5 singularity x^2y+y^4 on a genus-zero curve with three marked points, and I highlight two new phenomenons: non-concavity and the appearance of matrix factorizations. I give the definition of the virtual fundamental cycle in this special case and explain how to compute it. This illustrates a preliminary version of my main theorem and some of its ideas. At last, I state mirror symmetry for the D_5-singularity.

In the third lecture, I first explain the main motivations of the quantum singularity theory and the two main obstructions to its computation: non-concavity and the appearance of matrix factorizations. Then I focus on some particular class of singularities called chain, and I show how to rephrase in this case Polishchuk-Vaintrob construction of the virtual class without matrix factorizations. At last, I give a K-theoretic definition of the virtual class and state my main theorem on its computation. Also, I briefly explain Givental theory of the J-function and mirror symmetry.

In the last lecture, I give an extension of my main theorem in higher genus using the Hodge bundle and I give an analog of mirror symmetry in higher genus using the DR hierarchy introduced by Buryak. Then I prove my main theorem after introducing the notion of a recursive complex. It is a remarkable complex of vector bundles which benefits from several vanishing properties in cohomology. Finally, I will give some applications of the main theorem to tautological relations in the moduli space of stable curves.