## Topology Seminar (Main Talk): The symplectic $(A_\infty ,2)$-category

Seminar | November 7 | 4-5 p.m. | 3 Evans Hall

Nate Bottman, Princeton University

Department of Mathematics

In my second talk, I will describe a framework for building maps between Fukaya categories of different symplectic manifolds. This is a 2-category-like structure called Symp, where the objects are symplectic manifolds, the 1-morphisms are Lagrangians in products, and the 2-morphisms are intersections of these Lagrangians. Just as the structure of the Fukaya category comes from an operad of polytopes, the structure of Symp comes from a “relative 2-operad” of “2-associahedra”, which are new objects formulated recently by the speaker. I will highlight recent progress: a technique for computing composition maps in Symp in the context of symplectic quotients, and the definition of an $(A_\infty,2)$-category. Finally, I will mention work-in-progress with Katrin Wehrheim, which aims to complete the construction of Symp by formulating the relevant moduli spaces of quilts in terms of “family polyfolds”. This talk includes work joint with Shachar Carmeli and Katrin Wehrheim. There will be lots of pictures.

c_abbott@berkeley.edu