Positroid links and braid varieties

I will discuss a class of affine algebraic varieties associated with positive braids, their cluster structures and their relation to open Richardson varieties. First, I will explain our motivation which comes both from symplectic topology and from the study of HOMFLY-PT homology of torus links. Then we will discuss in detail how the study of  DG algebras associated with certain Legendrian links may help us to better understand the algebraic geometry of Richardson varieties. I will illustrate our results and conjectures concerning this interplay between topology and algebraic geometry with the example of open positroid varieties in Grassmannians. If time permits, I will briefly explain the relation between certain stratifications of braid varieties and cluster structures on their coordinate rings. This is joint work with Roger Casals, Eugene Gorsky, and José Simental.