Lagrangian caps in high dimensional symplectic manifolds I

Is it possible to find an embedded Lagrangian disk in \C^n - B^2n, so that the boundary is a Legendrian in S^{2n-1}? When n=2 the answer is no, but in all higher dimensions such disks exist in abundance. This follows from a more general existence theorem for Lagrangian embeddings with loose concave boundary; in this two-part talk we precisely state and prove this theorem. The proof has two main components: an action-balancing lemma for Lagrangian immersions, and a Lagrangian Whitney trick. We discuss the proof of both, in particular discussing how they both rely on the classification theorem for loose Legendrians. This project is joint work with Yakov Eliashberg.