Structure Constants for Bases of the Polynomial Ring

舒伯特微积分始于1879年左右，当时赫尔曼·舒伯特（Herman Schubert）提出几何中的枚举问题，并在特殊情况下回答这些问题。例如，平面上有多少个点同时与两条给定的直线相交，或者空间中有多少条直线与四条给定的直线相交？为了回答后一个问题，Schubert考虑了第一条线与第二条线相交，第三条线与第四条线相交的情况，在这种情况下，答案是2（连接两个交点的线和两对交点所跨越的两个平面的交点线）。然后他断言，根据他的数守恒原理，一般的答案，如果是有限的，也一定是2。大卫·希尔伯特，在他20世纪的第15个问题中，提出了严格的舒伯特数守恒原理的任务。上同调解决了这个问题，并将我们引向了现代舒伯特微积分和交理论，这些理论在几何、拓扑、组合学中都有分支，甚至在弦理论中也起着核心作用。然而今天，舒伯特最初的问题仍然没有得到回答：如何有效地计算线性子空间的交集数？Lascoux和Schutzenberger（1985）定义了完全旗簇上同调环上的Schubert类的多项式表示。这些舒伯特多项式给出了舒伯特类的显式多项式表示，其结构常数枚举舒伯特变种的合适三重相交中的标志（嵌套线性子空间序列），这正是舒伯特微积分旨在计算的非负整数。首席研究员将提升从对称函数理论到全多项式环的强大工具和技术，并将其应用于舒伯特多项式，最终目标是找到这些交点数的组合公式。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。

Schubert calculus began around 1879 with Herman Schubert asking, and in special cases answering, enumerative questions in geometry. For example, how many points in the plane meet two given lines simultaneously or how many lines in space meet four given lines? To answer the latter, Schubert considered the case where the first line intersects the second and the third intersects the fourth, in which case the answer is 2 (the line connecting the two points of intersection and the line of intersection of the two planes spanned by the two pairs of intersecting lines). He then asserted, by his principle of conservation of number, that the general answer, if finite, must also be 2. David Hilbert, in his 15th problem for the twentieth century, set out the task of making rigorous Schubert's principle of conservation of number. Cohomology resolved this and lead us to modern Schubert calculus and intersection theory, which has ramifications in geometry, topology, combinatorics, and even plays a central role in string theory. Yet today, Schubert's original question remains unanswered: how can one effectively compute intersection numbers for linear subspaces?Lascoux and Schutzenberger in 1985 defined polynomial representatives for the Schubert classes in the cohomology ring of the complete flag variety. These Schubert polynomials give explicit polynomial representations of the Schubert classes whose structure constants enumerate flags (sequences of nested linear subspaces) in a suitable triple intersection of Schubert varieties, which are precisely the nonnegative integers Schubert calculus aims to compute. The principal investigator will lift powerful tools and techniques from symmetric function theory to the full polynomial ring and apply them to Schubert polynomials with the ultimate goal of finding a combinatorial formula for these intersection numbers.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.