Lagrangian Cobordisms and Fukaya Categories

Unlike other cobordism categories, the category of Lagrangian
cobordisms inside a fixed symplectic manifold is stable.
This means it has, for instance, the structure of a triangulated category.
Moreover, we construct a functor to the Fukaya category
of that manifold respecting exact triangles. As a corollary, we will
see that Floer theory behaves similar to characteristic classes for
Lagrangian cobordisms. We'll also talk about a program for how
to show that the homotopy theory of Lagrangian cobordisms may
completely recover the Floer theory of the symplectic manifold.