Special Year Seminar I

We heard last week in Daoji's talk about positroid varieties, which are subvarieties in the Grassmannian defined by cyclic rank conditions, and which are related to Schubert varieties. In this talk, we will provide a criterion for whether positroid varieties are singular at certain distinguished points, and we will show that this information is sufficient to determine whether an entire positroid variety is smooth or singular. This will involve looking at combinatorial diagrams called "affine pipe dreams." We can also form a partial order on positroid varieties given by deletion and contraction, such that there is closure for smooth positroid varieties, and we will characterize the minimal singular elements in this order. Finally, we will discuss a couple applications to physics which involve the plabic diagrams introduced by Postnikov: the BCFW bridge decomposition and inverse soft factors.