Weave calculus and its applications

I will discuss progress in our understanding of Legendrian links in the standard Darboux 3-ball, focusing on their Lagrangian fillings. In particular, I will explain how to connect this symplectic geometric problem to the study of cluster algebras and use this connection to prove new results in both fields. This includes the construction of cluster structures for Richardson varieties for any simply-laced simple algebraic Lie group, addressing a conjecture of B. Leclerc, and hints towards an ADE classification for Lagrangian fillings. This is based on recent progress in the study of weaves and their microlocal aspects.