Flappy trees and monoidal structures on Fukaya categories

The SYZ version of Kontsevich's Homological Mirror Symmetry conjecture predicts that if X is a symplectic manifold with the structure of a (possibly singular) Lagrangian torus fibration and with a Lagrangian section, then the split-closed derived Fukaya category should be equivalent to the bounded derived category of coherent sheaves on a "dual" fibration. The latter category has a monoidal structure coming from the tensor product, which raises a natural question: in this geometric situation, does the Fukaya category carry a monoidal structure, ideally on the chain level? This was investigated in Aleksandar Subotic's 2010 thesis, in which Subotic enhanced D^pi Fuk(T^2) to a monoidal linear category and showed that Polishchuk--Zaslow's 1998 proof of HMS respects this monoidal structure. In joint work with Mohammed Abouzaid, we aim to establish Subotic's result in greater generality: in higher dimensions and in the presence of singularities, e.g. for a K3 surface. We also aim to construct this monoidal structure on the chain level. In this talk, I will describe how to make sense of the notion of a "monoidal A-infinity category" using "flappy trees".