Schubert Structure Constants via Kohnert Combinatorics

通过 Kohnert 组合学计算舒伯特结构常数

• 批准号: 2246785
• 负责人: Sami Assaf
• 金额: $ 21万
• 依托单位: University of Southern California
• 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=2246785&HistoricalAwards=false
• 关键词: Schubert; Structure; Constants; via; Kohnert

Classical enumerative geometry asks questions such as how many points lie in the intersection of two lines in the plane, or how many lines in space intersect four given lines? These enumeration questions counting linear subspaces satisfying certain geometric conditions have been considered since the late 1800s and have sparked many advances in mathematics including the development of modern intersection theory. Solutions to these enumeration questions connect to areas outside of mathematics through Gromov-Witten invariants and quantum cohomology. The formalism for solving such problems, known as Schubert calculus, is now rigorous, though the original question of enumerating linear subspaces remains largely unsolved. This project will develop simple combinatorial rules for computing these numbers. The vertically integrated research program will incorporate training for undergraduates, masters and doctoral students, and postdoctoral scholars. The first milestone is to develop rules in the cohomology ring of the classical flag manifold that occur in the product of an arbitrary Schubert class by one pulled back from a Grassmannian projection. The second milestone will utilize combinatorial crystal graphs to extend this rule to products of Demazure characters. The ultimate goal of this ambitious project is to generalize these two cases to give rules for the product of any two Schubert classes, thereby resolving the fundamental Schubert problem for the complete flag manifold.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.

经典的枚举几何形状提出了诸如飞机上两条线的交集中有多少点，还是空间中有多少行与四个给定线相交？这些枚举问题计算了满足某些几何条件的线性子空间，自1800年代后期以来，已经考虑了某些几何条件，并引发了数学的许多进步，包括现代相交理论的发展。这些枚举问题的解决方案通过格罗莫夫（Gromov）的不变性和量子共同体结合了数学以外的领域。解决此类问题的形式主义（称为Schubert conculus）现在很严格，尽管列举线性子空间的原始问题仍然在很大程度上无法解决。该项目将开发用于计算这些数字的简单组合规则。垂直整合的研究计划将纳入针对本科生，硕士和博士生的培训，以及博士后学者。第一个里程碑是在古典旗不同的舒伯特阶级产物中出现的经典国旗歧管的共同体学环中制定规则，从格拉曼尼亚的投影中撤回。第二个里程碑将利用组合晶体图将此规则扩展到扎唑字符的产品。这个雄心勃勃的项目的最终目标是概括这两个案例，以提供任何两个舒伯特课程的产品规则，从而解决了完整的旗手歧管的基本舒伯特问题。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛影响的评估来审查CRITERIA的评估。