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对象X施加一定的条件列表，那么存在多少个X，或者实际上根本就没有？任何这样的非线性问题（例如，“这条直线和那个圆上有多少个点？”可能有两个）都有线性化，甚至这些问题也很难回答。20世纪的代数工具足够强大，可以计算出基本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数减去另一个大数时，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消去的替代公式。本项目将培养研究生，指导博士后。这个项目包括三个子项目（几乎完全独立，尽管所有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交换的第二族算子。第二个子项目对一个古老的问题：“交换矩阵对的格式是简化格式吗？”PI将该方案退化为组件的并集（由“通用白日梦”索引，它们对自己的权利非常感兴趣），如果后一种并集被简化，那么交换方案也会被简化。第三个也是最后一个子项目介绍了一个用于计算“辫状变异”上同调的谱序列，它与链接的Khovanov同调、聚类代数和表示理论中的基本问题有关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.