Mathematics Colloquia and Seminars

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A number of spaces that naturally arise in algebra, combinatorics, and representation theory can be described as moduli spaces of flags or as configurations of points in a flag variety. Examples include Schubert varieties, Bott-Samelson varieties, and their many cousins. Such spaces seem to be better-behaved when we consider "polygons" of such flags, where we consider a cyclic sequence of flags and impose the condition that any two adjacent flags must be a certain distance apart. Why might this be the case?

To illustrate this, we'll focus on a simple class of such "polygons," called open brick manifolds. In spite of its simple definition, these turn out to encompass a surprising number of known spaces, including open Richardson varieties (i.e. an intersection of a Schubert variety with an opposite Schubert variety), positroid varieties, reduced double Bruhat cells, and certain spaces naturally associated with knots and braids, such as augmentation and braid varieties. By holding a mirror---literally, figuratively, and homologically---to such spaces, we are led to the existence of a number of "hidden" symmetries that must be satisfied: in its coordinate ring, in cohomology, and even in more subtle invariants, like its cluster algebra structure and motivic decomposition. We'll explain the general structures that allow for a uniform treatment and describe the explicit, combinatorial tools that allow us to make these abstractions concrete and bootstrap our intuition from familiar settings. This is joint work with A. Knutson.