Divided Differences, Pipe Dreams, Brick Manifolds, and Braid Varieties

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Many mathematical (and other-scientifical) problems are of the form: if we impose a certain list of conditions on some desired object X, how many Xs exist, or indeed are there any at all? Any nonlinear such problem (for example, "how many points lie both on this line and that circle?" probably two) has linearizations and even those can be hard to answer. 20th century algebraic tools are powerful enough to compute the number of Xs satisfying basically any such linearized problem; however, when that number is computed as one big sum minus another big sum, it can be hard to predict when the result is zero vs. positive! Much of the PI's work, and the first part of this project, is in a search for alternate formulae that do not involve any cancelation. Graduate students will be trained and postdocs will be mentored during the course of this project.This project consists of three subprojects (almost wholly independent, though all straddle algebraic combinatorics and algebraic geometry). The first subproject concerns the divided-difference-operator recurrence that defines Schubert polynomials, characterizing and exploiting a second family of operators that commute with the first. The second subproject has a new approach to an age-old question: "is the scheme of pairs of commuting matrices a reduced scheme?" The PI has a degeneration of this scheme to a union of components (indexed by "generic pipe dreams", which are of great interest in their own right), and if this latter union is reduced then so too is the commuting scheme. The third and final subproject introduces a spectral sequence for computing the cohomology of "braid varieties", which connects to Khovanov homology of links, to cluster algebras, and to basic questions in representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.