Schubert Structure Constants via Kohnert Combinatorics

经典的枚举几何会问这样的问题：平面上两条直线的交点上有多少个点，或者空间中有多少条直线与四条给定的直线相交？这些计数问题计数线性子空间满足一定的几何条件已被认为是自1800年代后期，并引发了许多数学进步，包括现代交叉理论的发展。这些枚举问题的解决方案通过Gromov-Witten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（即舒伯特微积分）现在已经很严格，尽管最初的线性子空间枚举问题在很大程度上仍未解决。这个项目将开发计算这些数字的简单组合规则。垂直整合的研究计划将包括本科生，硕士和博士生，以及博士后学者的培训。第一个里程碑是制定规则的上同调环的经典旗流形，发生在产品的任意舒伯特类拉回到一个格拉斯曼投影。第二个里程碑将利用组合晶体图将这一规则扩展到Demazure字符的产品。这个雄心勃勃的项目的最终目标是推广这两种情况下给出任何两个舒伯特类的产品的规则，从而解决了基本的舒伯特问题的完整的旗流形。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.