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.

经典的枚举几何提出了一些问题，例如平面中两条线的交点有多少点，或者空间中有多少条线与四条给定线相交？这些计算满足某些几何条件的线性子空间的枚举问题自 1800 年代末以来一直被考虑，并引发了数学的许多进步，包括现代交集理论的发展。这些枚举问题的解决方案通过格罗莫夫-维滕不变量和量子上同调连接到数学之外的领域。解决此类问题的形式主义（称为舒伯特微积分）现在是严格的，尽管枚举线性子空间的原始问题在很大程度上仍未得到解决。该项目将开发简单的组合规则来计算这些数字。垂直整合的研究计划将包括对本科生、硕士生、博士生以及博士后学者的培训。第一个里程碑是开发经典标志流形的上同调环中的规则，这些规则出现在从格拉斯曼投影拉回的任意舒伯特类的乘积中。第二个里程碑将利用组合晶体图将此规则扩展到 Demazure 角色的产品。这个雄心勃勃的项目的最终目标是概括这两种情况，为任意两个舒伯特类的乘积给出规则，从而解决完整旗形流形的基本舒伯特问题。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。