Schubert Structure Constants via Kohnert Combinatorics

经典的枚举几何提出了这样的问题：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线相交？这些计算满足某些几何条件的线性子空间的计数问题自19世纪末以来一直被考虑，并引发了许多数学上的进步，包括现代交集理论的发展。这些枚举问题的解决方案通过Gromov-Witten不变量和量子上同调联系到数学之外的领域。解决这类问题的形式主义，称为舒伯特微积分，现在是严格的，尽管最初的列举线性子空间的问题在很大程度上仍然没有解决。这个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、硕士和博士后以及博士后学者的培训。第一个里程碑是发展经典旗形的上同调环中的规则，这些规则出现在任意Schubert类与从Grassman投影拉回的乘积的乘积中。第二个里程碑将利用组合水晶图将这一规则扩展到Demazure角色的产品。这个雄心勃勃的项目的最终目标是推广这两个案例，为任何两个舒伯特类别的乘积提供规则，从而解决完整旗帜流形的根本舒伯特问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.