Combinatorics of Sharing Theorems, Stratifications, Bruhat Theory and Shimura Varieties

共享定理、分层、Bruhat 理论和 Shimura 簇的组合

• 批准号: 2247382
• 负责人: Margaret Readdy
• 金额: $ 18.27万
• 依托单位: University of Kentucky Research Foundation
• 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: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247382&HistoricalAwards=false
• 关键词: Combinatorics; Sharing; Theorems; Stratifications; Bruhat

This project is jointly funded by the Combinatorics program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Algebra and Number Theory program. Combinatorics is the area of mathematics concerned with enumerating, understanding and characterizing mathematical structures that occur within discrete objects. The results of this project will contribute to the growing connections between combinatorics and other mathematical disciplines. This includes extending sharing theorems of classical Coxeter groups, using combinatorial methods to generalize and understand identities involved with the representation theory of Shimura varieties, and continuing the PI's successful program involving noncommutative polynomials that have applications to understanding polytopes, Whitney stratified spaces and the totally nonnegative flag variety. Results from this award have the potential to give insight into discrete structures in other sciences including the noncommutative nature of DNA sequencing in biology, mathematical methods in coding theory and topological surfaces related to robotic motion. The PI has been vigorously involved in research, educational and outreach activities to foster the growth of the mathematical workforce. As a response to the isolation caused by the COVID-19 pandemic, this includes co-organizing a new regional lecture series to reinvigorate collaboration between students, postdocs and faculty. Support of the PI's research program reinforces the NSF's goals of scientific progress, building new talent, fostering innovation and improving society.More specifically, this project includes four subprojects involving combinatorics broadly defined, plus ongoing graduate research projects. Results from the award will contribute to the growing connections between combinatorics with geometry, topology, algebra and number theory, and to make fundamental contributions to classical areas of combinatorics. Project I involves the PI's new work with Ehrenborg and Morel on extensions of sharing theorems to Coxeter arrangements, and using Herb's theory of 2-structures to give dissection proofs and generalizations to intrinsic volumes. Extensions to other geometric settings will be studied, including regular complex polytopes and the Nandakumar--Rao conjecture. Project II involves the PI's joint work with Ehrenborg and Goresky on topological face enumeration of Whitney stratified spaces, more generally, zeta functions of quasi-graded posets. This widens the research program to understand and make progress on face vector inequalities for polytopes and singular spaces. The PI and Ehrenborg develop a non-homogeneous extension of the classical cd-index to labeled digraphs satisfying a balanced condition to generalize the setting of Eulerian graded posets and include the family of Bruhat graphs as a special case. Project III includes a conjecture for balanced digraphs which implies the Billera--Brenti nonnegativity conjecture for the cd-index of Bruhat graphs. Project IV concerns extending the combinatorics of the generalized Harish-Chandra character formula. The PI will continue her educational, regional and national activities to support the long-range goal to attract, retain and train more under-represented groups in the mathematical sciences and ultimately increase the number of STEM-educated individuals in the workforce.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.

该项目由组合学项目、刺激竞争研究的既定项目（EPSCoR）以及代数和数论项目共同资助。组合学是数学的一个领域，涉及枚举、理解和描述发生在离散对象中的数学结构。这个项目的结果将有助于组合学和其他数学学科之间日益增长的联系。这包括扩展经典Coxeter群的共享定理，使用组合方法推广和理解与Shimura变体表示理论有关的恒等式，并继续PI的成功计划，该计划涉及应用于理解多面体，Whitney分层空间和完全非负标志变体的非交换多项式。该奖项的结果有可能为其他科学中的离散结构提供见解，包括生物学中DNA测序的非交换性质，编码理论中的数学方法以及与机器人运动相关的拓扑表面。PI一直积极参与研究，教育和推广活动，以促进数学劳动力的增长。作为对COVID-19大流行造成的隔离的回应，这包括共同组织一个新的区域系列讲座，以重振学生、博士后和教师之间的合作。对PI研究项目的支持加强了NSF在科学进步、培养新人才、促进创新和改善社会方面的目标。更具体地说，该项目包括四个子项目，涉及广泛定义的组合学，以及正在进行的研究生研究项目。该奖项的结果将有助于组合学与几何、拓扑、代数和数论之间日益增长的联系，并对组合学的经典领域做出根本性的贡献。项目一涉及PI与Ehrenborg和Morel的新工作，将共享定理扩展到Coxeter排列，并使用Herb的2-结构理论给出解剖证明和对内量的推广。扩展到其他几何设置将被研究，包括正则复杂多面体和Nandakumar- Rao猜想。项目II涉及PI与Ehrenborg和Goresky在Whitney分层空间的拓扑面枚举方面的联合工作，更一般地说，是准梯度偏序集的zeta函数。这扩大了对多面体和奇异空间的面向量不等式的理解和研究。PI和Ehrenborg将经典的cd-索引推广到满足平衡条件的标记有向图，推广了欧拉梯度偏序集的集合，并将Bruhat图族作为特例纳入其中。方案三包括平衡有向图的一个猜想，该猜想暗示了Bruhat图cd-index的Billera—Brenti非负猜想。项目四涉及扩展广义Harish-Chandra字符公式的组合学。PI将继续她的教育、区域和国家活动，以支持吸引、留住和培训更多代表性不足的数学科学群体的长期目标，并最终增加劳动力中受过stem教育的个人的数量。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。