Abstract: The Fukaya category is a powerful invariant of symplectic manifolds, which plays a key role in homological mirror symmetry. I will begin by describing my program to relate the Fukaya categories of different symplectic manifolds. The key objects are “witch balls” (coupled systems of PDEs whose domain is the Riemann sphere decorated with circles and points), as well as the configuration spaces of these domains, which are posets called “2-associahedra”. Next, I will describe associated work-in-progress in which I construct cellular decompositions of the constituent spaces of the Fulton–MacPherson operad. The k-th space in this operad is a compactification of the space of configurations of k labeled points in R^2, up to translations and dilations, and the approach we take is related to Getzler-Jones’ attempted decomposition from 1994.