An interesting problem in contact topology is to understand the Lagrangian surfaces that fill a given Legendrian link in a contact 3-manifold. A key breakthrough this year is that we now know some families of Legendrian links that have infinitely many different fillings. This is due to various work by Honghao Gao, Linhui Shen, Daping Weng, and some people who aren't currently at Michigan State, and the common approach is through microlocal sheaf theory and clusters. I'll describe a different, Floer-theoretic approach to the same sort of result, using augmentations of Legendrian contact homology. The Floer approach recovers some of the results from the sheaf approach and also produces some new examples of links with infinitely many fillings. This is joint work in progress with Roger Casals.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09