Speaker: Yunpeng Niu (Stony Brook University) 

Title: A Monoidal Structure on the Fukaya Category

Abstract:  Homological mirror symmetry suggests that there should exist on the Fukaya category certain structures analogous to those known on the derived category of coherent sheaves. In 2010, A. Subotic associated a natural monoidal structure on the generalized Donaldson-Fukaya category, mirror to the tensor product of coherent sheaves, when the symplectic manifold is equipped with a smooth Lagrangian fibration with a distinguished Lagrangian section. In this talk, I will describe a generalization of this result: how given a Lagrangian fibration (with section) with focus-focus type singularities on a symplectic 4-fold M, one can associate a natural immersed Lagrangian correspondence from M x M to M. In the case of a generic elliptic K3, I will explain why such a correspondence is unobstructed and how it induces a monoidal structure on the Fukaya category of M. This is part of joint work with M. Abouzaid and N. Bottman.