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：如果我们对某个期望的对象X强加一个特定的条件列表，那么存在多少个X，或者实际上根本就没有X？任何这样的非线性问题(例如，“这条线上和那个圆上都有几个点？”可能有两个)有线性化，即使是线性化也很难回答。20世纪的代数工具足够强大，可以计算出基本上满足任何这种线性化问题的X的数量；然而，当这个数字被计算为一个大和减去另一个大和时，很难预测结果是零还是正！PI的大部分工作，也是这个项目的第一部分，是寻找不涉及任何抵消的替代公式。在这个项目的过程中，将培训研究生和指导博士后。这个项目由三个子项目组成(几乎完全独立，尽管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，刻画和利用了与第一个算子交换的第二族算子。第二个子项目有一个新的方法来解决一个古老的问题：“交换矩阵对的方案是简化方案吗？”PI将这一方案退化为一个组件联盟(由对其本身非常感兴趣的“通用白日梦”索引)，如果后者的联盟被减少，那么通勤计划也是如此。第三个也是最后一个子项目介绍了一个用于计算“辫子簇”的上同调的谱序列，它与链环的Khovanov同调、簇代数和表示理论中的基本问题有关。这个奖项反映了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，被认为是值得支持的。

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.