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

许多数学（和其他科学）问题的形式如下：如果我们对某个所需的对象 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？任何非线性的此类问题（例如，“这条线和那个圆上有多少点？”可能有两个）都有线性化，甚至这些也很难回答。 20 世纪的代数工具足够强大，可以计算基本上满足任何此类线性化问题的 X 的数量；然而，当该数字被计算为一个大和减去另一个大和时，很难预测结果何时为零或正！ PI 的大部分工作以及该项目的第一部分是寻找不涉及任何取消的替代公式。在该项目过程中，研究生将接受培训，博士后将得到指导。该项目由三个子项目组成（几乎完全独立，尽管所有子项目都跨越代数组合学和代数几何）。第一个子项目涉及定义舒伯特多项式的除差算子递归，描述和利用与第一个算子可交换的第二个算子系列。第二个子项目对一个古老的问题采用了新方法：“通勤矩阵对的方案是简化方案吗？” PI 将该方案退化为组件的联合（由“通用白日梦”索引，这些梦想本身就很有趣），如果后一个联合减少，那么通勤方案也会减少。第三个也是最后一个子项目介绍了用于计算“辫子簇”上同调的谱序列，该序列与链接的霍瓦诺夫同调、聚类代数以及表示论中的基本问题相关。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.