Nathaniel Bottman

I will present the 2-associahedra, which I constructed in 2017 in the context of symplectic geometry, and explain some current developments, focusing on symplectic aspects. First, I will explain how 2-associahedra form the right operadic structure for endowing the Fukaya category — the most important invariant of a symplectic manifold — with functoriality properties. Specifically, the 2-associahedra lead to the notion of the symplectic (A-infinity,2)-category Symp, which is the natural setting for building functors, associated to Lagrangian correspondences, between Fukaya categories. Second, I will describe a related family of posets called constrainahedra, which Daria Poliakova and I constructed in 2022. The constrainahedra also translate into an algebraic structure in symplectic geometry: in ongoing work with Mohammed Abouzaid and Yunpeng Niu, we aim to show that Fukaya categories of Lagrangian torus fibrations are monoidal A-infinity categories, where the latter notion is constructed using of constrainahedra. This will fulfill a longstanding expectation in mirror symmetry. I will not assume a symplectic background.