Title: Legendrian invariants, Lagrangian fillings and cluster algebras

 Abstract: Classifications of Legendrian knots and their exact Lagrangian fillings are central questions in low-dimensional contact and symplectic topology. Recent development suggests that one can use cluster seeds to distinguish exact Lagrangian fillings. It requires a filling-to-cluster functoriality over a moduli space of Legendrian invariants. This invariant can be sheaf theoretic (Shende-Treumann-Williams-Zaslow) or Floer theoretic (joint work with Linhui Shen and Daping 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 Roger Casals, using sheaf-theoretic invariant), and for most positive braid links (joint work with Linhui Shen and Daping Weng, using Floer-theoretic invariant).