Combinatorial and algebraic methods in Schubert geometry

PI将研究有关Grassmannians和FLAG流形中Schubert簇的几何的组合、代数和计算问题。这两个主要项目探讨了该主题中的核心问题。第一个项目寻求在舒伯特微积分中统一表述的组合规则，通过杨图景的扩展理论。著名的Littlewood-Richardson系数就是在这一背景下产生的，但也出现在一般线性群和对称群的表示理论、Hermite矩阵和的特征值以及有限交换p-群的短正合序列的理论中。PI将在与H.Thomas共同工作的基础上，为这些系数及其舒伯特微积分扩展找到组合规则。第二个项目旨在进一步开发一个与A.Woo共同介绍的组合和计算框架，以了解Schubert变分的奇点。该项目中应用和进一步开发的主要工具来自组合学，包括代数和几何组合学，以及组合交换代数。组合学涉及离散的对象，如排列、偏序和图，以及枚举技术。在许多情况下，例如在舒伯特变种中，连续对象可以通过离散数据来参数化，从而为组合分析打开了大门。此外，人们经常看到，相同的组合对象先验地支配着不同的数学设置。利特尔伍德-理查森系数就是一个很好的例子。该项目同样寻求在数学领域之间以及与其他科学学科之间进一步寻找思想的交叉流动。

The PI will study combinatorial, algebraic and computational questions about the geometry of Schubert varieties in Grassmannians and flag manifolds. The two main projects explore central problems in the subject. The first project seeks uniformly stated combinatorial rules in Schubert calculus, via an expanded theory of Young tableaux. The famous Littlewood-Richardson coefficients arise in this context, but also in the theory of representations of general linear groups and of symmetric groups, eigenvalues of sums of Hermitian matrices and short exact sequences of finite abelian p-groups. The PI will find combinatorial rules for these coefficients and their Schubert calculus extensions, building on joint work with H. Thomas. The second project aims to further develop a combinatorial and computational framework, introduced jointly with A. Woo, to understand the singularities of Schubert varieties.The main tools applied and further developed in this project come from combinatorics, including algebraic and geometric combinatorics, and combinatorial commutative algebra. Combinatorics concerns discrete objects such as permutations, partial orders and graphs, as well as techniques of enumeration. In many instances, such as with Schubert varieties, continuous objects can be parameterized by discrete data, opening the door for combinatorial analysis. Moreover, one often witnesses that the same combinatorial objects govern a priori different mathematical settings. The Littlewood-Richardson coefficients are a prime example of this. The project seeks to similarly find further cross-flow of ideas between areas of mathematics, as well as with other scientific disciplines.