Cost of splitting Lagrangians

Assume that the derived Fukaya category of a symplectic manifold admits a collection of triangular generators. By definition, this means that any other Lagrangian submanifold which is an object of this category can be decomposed in terms of exact triangles involving the generators. The purpose of the talk is to explain why such a decomposition requires a certain non-trivial amount of “energy”. The notion of energy that appears here is an extension of Hofer's energy. A first part of the argument consists in bounding from below the shadow of multiple ended Lagrangian cobordisms and will be discussed in some detail in the talk. I will only be sketch a second part that remains, for technical reasons, somewhat speculative at this time. This involves translating the derived Fukaya category into a cobordism category with objects and morphisms immersed Lagrangians. The talk is based on joint work in progress with Paul Biran and Egor Shelukhin.