Combinatorial and algebraic methods in Schubert geometry

PI将研究关于Grassmannians和旗流形中舒伯特簇的几何的组合，代数和计算问题。这两个主要项目探讨了该主题的核心问题。第一个项目寻求统一的舒伯特微积分的组合规则，通过扩展的理论杨tableaux。著名的Littlewood-Richardson系数出现在这方面，但也在一般线性群和对称群的表示理论中，埃尔米特矩阵和有限阿贝尔p群的短精确序列的特征值。PI将在与H的联合工作的基础上，为这些系数及其舒伯特微积分扩展找到组合规则。托马斯。第二个项目旨在进一步发展一个组合和计算框架，与A。Woo，理解Schubert簇的奇异性。本项目应用和进一步发展的主要工具来自组合数学，包括代数和几何组合数学，以及组合交换代数。组合学涉及离散对象，例如排列、偏序和图，以及枚举技术。在许多情况下，例如舒伯特变种，连续对象可以由离散数据参数化，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。Littlewood-Richardson系数就是一个很好的例子。该项目旨在类似地寻找数学领域之间以及与其他科学学科之间的思想交流。

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.