Daniil Mamaev – Defining Fukaya Categories Geometrically

Abstract: A Fukaya category F(M) of a symplectic manifold M is supposed to be a triangulated A-infinity category whose objects are (some) Lagrangians (with additional structure), morphism spaces are given by intersection points, and A-infinity operations are given by (virtual) counts of some pseudoholomorphic polygons with sides on Lagrangians and corners at their intersection points. Implementations vary, but usually F(M) is obtained from a category depending on choices via an algebraic gadget such as triangulated envelope or A-infinity quotient. I will explain, without assuming any familiarity with Fukaya categories, how to get a triangulated category directly from geometric data in the case of relative wrapped Fukaya categories of surfaces. If time permits, I will also outline how these categories arise in homological mirror symmetry.