Monoidal Structures on Fukaya Categories

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. Given a Lagrangian fibration with focus-focus type singularities on a symplectic 4-fold $M$, I will describe how to associate geometrically an immersed Lagrangian correspondence from $M \times M$ to $M$; in the case of a generic elliptic K3, this correspondence yields an $A_\infty$ bi-functor on its Fukaya category, which is expected to give rise to a monoidal structure. This is partially based on joint work with Mohammed Abouzaid and Nathaniel Bottman.