Categorical symplectic geometry studies an invariant of symplectic manifolds called the "Fukaya (A-infinity) category", which consists of the Lagrangian submanifolds and a symplectically-robust intersection theory of these Lagrangians. Over the last two decades the Fukaya category has emerged as a powerful tool: for instance, it has produced inroads to Arnol'd's Nearby Lagrangians Conjecture, and it allowed Kontsevich to formulate the the Homological Mirror Symmetry conjecture.
In this talk I will describe a project, joint with Satyan Devadoss, Stefan Forcey, and Katrin Wehrheim, which attempts to relate the Fukaya categories of different symplectic manifolds via a notion of functoriality. After mentioning some analytic results about the singular quilts necessary for this construction, I will describe the combinatorial component: with Devadoss and Forcey we are constructing a family of polytopes that specialize to the associahedra in two different ways, and can be thought of as the 2-categorical version of associahedra.
- Mathematical physics
- Scientific Series