Title: Enumerative geometry of curves from GIT quotients
Abstract: Gromov-Witten theory of toric varieties is known in any genus. It is not the case for projective hypersurfaces in positive genus, and even in genus zero, it is not known for hypersurfaces in weighted projective spaces. To try to answer this difficult question, Fan, Jarvis, and Ruan have moved to another point of view: they see the polynomial defining the hypersurface as an orbifold singularity and they defined an analogue of Gromov-Witten theory for it. In this talk, I will describe a joint work with Ciocan-Fontanine, Favero, Kim, and Shoemaker whose goal is to define an analogue of Gromov-Witten theory of complete intersections in toric varieties. The construction is based on matrix factorizations and gives a new point of view on Gromov-Witten theory.