| Abstract: To a positive braid $\beta$ on $n$ strands we associate an affine algebraic variety $X(\beta)$ as the solution space of an incidence problem in the flag variety of $\operatorname{GL}_{n}(\mathbb{C})$. This variety has a number of nice properties, for example, it is smooth and admits smooth compactifications which correspond to different braid words for $\beta$. Many interesting varieties appearing in Lie theory, such as open Richardson varieties in type $A$, appear as varieties of the form $X(\beta)$ for special types of braids $\beta$. I will define these varieties and explain some of their combinatorial and geometric properties. Most importantly, I will describe an $\mathcal{A}$-cluster structure on $X(\beta)$ defined using the formalism of algebraic weaves, a graphical calculus that allows us to find open tori in $X(\beta)$. In particular, this yields a cluster structure on type $A$ Richardson varieties. Finally and time permitting, I will elaborate on the relationship that $X(\beta)$ bears to the Legendrian link appearing as the Legendrian $(-1)$-closure of the braid $\beta$, and how a cluster structure on $X(\beta)$ may help in the understanding of the symplectic geometry of this link. This talk is based on joint works with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le and Linhui Shen. |