Fan-Jarvis-Ruan theory, part I

I will describe recent joint work with Arkady Vaintrob. Fan-Jarvis-Ruan theory is an analog of Gromov-Witten theory in which a target space is replaced by a quasihomogeneous isolated hypersurface singularity. In the case of simple singularities of type A it corresponds to the intersection theory on the moduli space of higher spin curves, and constitutes a framework for the famous Witten conjectures, proved by Kontsevich for A$_1$-singularity and by Faber-Shadrin-Zvonkine in general. In my lectures I will explain the algebro-geometric construction of the relevant cohomological field theory based on the theory of matrix factorizations. The crucial construction of an analog of the virtual fundamental class involves derived categories of matrix factorizations in a global setting.