Lagrangian submanifolds: from the local model to the cluster complex

Abstract

In these notes, I will present collaborative works on Lagrangian submanifolds that I realised mainly with Octav Cornea (the cluster complex, which naturally leads to a universal Lagrangian Floer theory), but also, at an earlier stage, with Jean-Claude Sikorav. To cover the subject in a more complete and adequate way, I will also mention very recent works by Barraud–Cornea and by Welschinger, closely related to the subject of these notes. The aim of the cluster machinery is to resolve the well known problem of real codimension 1 bubbling off of disks in the Gromov–Floer theory; see Fukaya–Oh–Ohta–Ono (especially the two lectures by Oh and Ono in these proceedings) for a different, earlier, approach.