The completion of the set of Lagrangians and applications to dynamics -- Based on lectures by C. Viterbo

The goal of these lectures is to introduce the completion of the set of Lagrangian submanifolds of a symplectic manifold with respect to the spectral metric first introduced by V. Humilière and recently revisited by C. Viterbo. We establish a number of basic properties of this completion, in particular through the notion of $γ$-support, which we develop as a refinement of Humilière's original concept. We then present an application of these notions to conformally symplectic dynamics, generalizing the notion of Birkhoff attractor as defined and studied by G.D. Birkhoff, M. Charpentier, and more recently P. Le Calvez. Finally, we briefly mention several other applications of the Humilière completion and highlight many open questions. These are notes elaborated from the lectures with the same title given by C. Viterbo at the CIME School ''Symplectic Dynamics and Topology'' held in Cetraro (CS), Italy, from 16th to 20th June 2025.