Approximability for Lagrangian Submanifolds

In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups and as a generalization of total-boundedness. I will explain how recent technological advances in symplectic topology and persistence category theory allow us to talk about approximablity of spaces of Lagrangian submanifolds and discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.