I will discuss how many exact Lagrangian fillings of Legendrian torus knots are possible up to Hamiltonian isotopies.
The main tools are augmentations of Legendrian DGA and Lagrangian saddle cobordisms.
I will also talk about how the cluster structure and Dynkin diagrams are related to this subject.
This is a joint work in progress with Byunghee An.

Lecture 3: Cluster structures and space of augmentations.