Wednesday, November 4th, 2015
03:00pm - 04:00pm, in McCormack 2-205

Nate Bottman

Northeastern University

Polytopes and categorical symplectic geometry

Abstract: Categorical symplectic geometry studies an invariant of symplectic manifolds called the ``Fukaya (\(A_\infty\)) 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 a bridge between symplectic and algebraic geometry called the Homological Mirror Symmetry conjecture. In this talk I will explain how a family of polytopes called the ``associahedra'' relate to the Fukaya category, and how the combinatorics of associahedra lead to the particular algebraic flavor of the Fukaya category. Next, I will describe a project that is joint with Satyan Devadoss, Stefan Forcey, and Katrin Wehrheim, that attempts to relate the Fukaya categories of different symplectic manifolds via a notion of functoriality. I will sidestep the formidable analytic aspects of this project and focus on 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. This talk will be accessible to all mathematicians, and there will be plenty of pictures.