Infinitely many Lagrangian fillings

An interesting problem in contact topology is to understand the Lagrangian surfaces that ”fill” a given Legendrian knot or link in a contact 3-manifold. A key breakthrough in the past year or so has been the discovery that some families of Legendrian links have infinitely many different fillings. There are now a variety of approaches to proving results along these lines, using techniques from microlocal sheaf theory, cluster algebras, and Floer theory. I’ll focus on this last approach, and describe a concrete way to construct Legendrian links with infinitely many fillings that can be distinguished using Legendrian contact homology. This is joint work with Roger Casals.