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

许多数学（和其他科学）问题都是这样的形式：如果我们对某个期望的对象X施加一定的条件列表，那么存在多少个X，或者确实存在任何X？任何非线性问题（例如，“这条直线和那条圆上有多少个点？“可能是两个）具有线性化，即使是那些也很难回答。20世纪世纪的代数工具已经足够强大，可以计算出基本上满足任何线性化问题的X的数量;然而，当这个数字被计算为一个大的总和减去另一个大的总和时，很难预测结果是零还是正！PI的大部分工作，以及这个项目的第一部分，都是在寻找不涉及任何取消的替代公式。本项目包括三个子项目（几乎完全独立，但都跨越代数组合学和代数几何学）。第一个子项目涉及的除差算子递归定义舒伯特多项式，特征和利用第二个家庭的运营商，第一个交换。第二个子项目对一个古老的问题有一个新的方法：“交换矩阵对的方案是一个简化方案吗？”PI将这个方案退化为一个组件的联合（由“通用的白日梦”索引，它们本身就很有兴趣），如果后者的联合被减少，那么通勤方案也是如此。第三个也是最后一个子项目介绍了一个光谱序列计算的“辫子品种”，连接到Khovanov同源的链接，集群代数，并在表示论的基本问题的上同调。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.