Exact Lagrangian fillings and clusters
A general belief is that exact Lagrangian fillings can be distinguished using cluster theory. In this talk, I will present such a framework via Floer theory — given a positive braid Legendrian link, its augmentation variety is a cluster K_2 variety and its admissible fillings induce cluster seeds (joint work with L. Shen and D. Weng).
As an application, I will explain how to use Legendrian loops and cluster algebras to construct infinitely many exact Lagrangian fillings for most torus links (joint work with R. Casals, using sheaves in accordance with Shende-Treumann-Williams-Zaslow), and for most positive braid links (joint work with L. Shen and D. Weng, using augmentations). Time permitting, I will also survey other methods to produce infinitely many fillings and compare these approaches.