Points and lines

The Fukaya category of a symplectic manifold is a robust intersection theory of its Lagrangian submanifolds. Over the past decade, ideas emerging from Wehrheim--Woodward's theory of quilts have suggested a method for producing maps between the Fukaya categories of different symplectic manifolds. I have proposed that one should consider maps controlled by compactified moduli spaces of marked parallel lines in the plane, called "2-associahedra". In this talk I will describe the 2-associahedra, with a focus on their topological and combinatorial aspects; in particular, I will produce combinatorial data that are in bijection with the strata of the 2-associahedra, and describe a generating function technique for computing the number of dimension-$m$ strata in a particular 2-associahedron. I will not assume any familiarity with symplectic geometry.