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

分歧、白日梦、砖流形和辫子品种

• 批准号: 2246959
• 负责人: Allen Knutson
• 金额: $ 36万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246959&HistoricalAwards=false
• 关键词: Divided; Differences; Pipe; Dreams; Brick

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.

许多数学（和其他科学）问题的形式是：如果我们对某些所需的对象X强加一定的条件列表，则存在多少X，或者实际上有任何条件？任何非线性的问题（例如，“这条线和那个圈子上有多少个点？大概是两个），也很难回答。 20世纪代数工具足够强大，可以计算满足所有此类线性问题的XS数量；但是，当将该数字计算为一个大总和减去另一个大和总和时，很难预测结果何时为零与正值！ PI的大部分工作以及该项目的第一部分都是在寻找不涉及任何取消的替代公式。研究生将接受培训，并且将在该项目过程中进行指导。该项目由三个副投影（几乎完全独立，尽管所有跨越代数组合术和代数几何形状）组成。第一个子项目涉及定义Schubert多项式的分裂差异 - 操作员的复发，表征和利用了第二个使用第一个通勤的操作员。第二个子项目对一个古老的问题有了新的方法：“通勤矩阵的方案是否减少了方案？” PI将该计划的退化为组成的结合（由“通用的烟斗梦”索引，这本身就是极大的兴趣），如果后一个联盟减少了，那么通勤计划也是如此。第三个也是最后一个子项目引入了一个用于计算“编织品种”的共同体的光谱序列，该序列与链接的Khovanov同源性连接到群集代数，以及代表理论中的基本问题。该奖项奖反映了NSF的法定任务，并通过评估了基金会的Merit和Broadial and crowia and crowia and Broadia and Broadia and Broadia and Broadia and crowia and tocriatial and Broadia and tocrita和broaditial and crowia and tocriatia and tobroit and tobrit and tobrit and tobrit and tobrit and tobrit。