Braid varieties, torus actions and stratifications

I will discuss braid varieties, a class of affine algebraic varieties associated to positive braids and related to augmentation varieties and open Bott-Samelson varieties. First, I will explain the geometric properties of braid varieties, including the construction of torus actions and holomorphic symplectic structures on their quotients. Then, we will discuss correspondences between these braid varieties constructed by using moduli spaces associated to certain labeled planar diagrams. I will explain how these geometric correspondences induce stratifications for braid varieties and their quotients, unifying known constructions of A. Mellit, in the case of character varieties, and M. Henry and D. Rutherford, in the case of augmentation varieties. If time permits, I will briefly explain the relation between open toric charts appearing in these stratifications and cluster algebras. (Based on joint work with R. Casal, E. Gorsky, and J. Simental.)