Discrete Geometry and Combinatorics Seminar
I will explain my construction of the 2-associahedra, which are graded posets indexed by sequences of nonnegative integers. The 2-associahedra arose in symplectic geometry, where they control functoriality for a symplectic invariant called the Fukaya category. The elements of a 2-associahedron correspond to the degenerations in the compactified configuration space of marked points on vertical lines in R^2, up to translations and positive dilations. I will explain several properties of the 2-associahedra: they are abstract polytopes (in particular, they are thin and strongly connected); they are Eulerian lattices (joint with my student Dylan Mavrides); and they have an operad-like structure that makes it possible to compute the (flag)-f-vectors using generating function techniques. I will discuss the conjecture that the 2-associahedra can be realized as convex polytopes. Finally, I will mention ongoing work with Alexei Oblomkov, in which we cast the topological realizations of 2-associahedra as an instance of a version of Fulton-MacPherson compactification for a pair of spaces. This will be a combinatorics talk; in particular, I will assume no symplectic background.