Approximability for Lagrangians

In recent joint work with Paul Biran and Octav Cornea, we introduced a notion of approximability for metric spaces whose points are objects of a triangulated category. This provides a quantitative analogue of (split-)generation. I will explain in what sense certain Fukaya categories (for instance, closed exact Lagrangians in cotangent bundles) are approximable and discuss applications to the rigidity and complexity of Lagrangian submanifolds. If time permits I will discuss how this story is realated to recent development in Lagrangian topology.