Schubert varieties: Combinatorics, Computation and Geometry

提案者将在格拉斯曼、旗流形及其舒伯特变种的组合学、几何和拓扑结构方面进行一系列主要项目的合作。这些变体是跨越经典数学许多领域的问题的核心，例如代数拓扑、枚举几何和表示理论。他们希望解决一些关于奇点分类和Schubert变元拓扑的棘手问题，例如30年的格拉斯曼不动点性质问题。他们还提出了几个额外的项目，继续PI和co-PI过去在代数组合方面的工作。这些项目涉及到最近发现的“循环筛分现象”，涉及到Coxeter群的交替子群，以及图的关键群。其中一些项目是PI过去（nsf资助）REU的产物。这项工作的吸引力在于，计数问题是组合学领域的主要问题之一，看起来与代数或几何无关，有时可以在这些其他领域找到解决和/或应用的关键。这对于本提案中的代数/几何对象尤其如此：旗流形和格拉斯曼流形中的舒伯特变体。这些物体是理解线和平面如何在二维平面、三维空间和高维空间中以不同角度倾斜和相交的关键。

The proposers will collaborate on a suite of main projects in the combinatorics, geometry, and topology of Grassmannians, flag manifolds, and their Schubert varieties. These varieties are central to questions spanning many areas of classical mathematics, such as algebraic topology, enumerative geometry, and representation theory. They hope to resolve some thorny questions about classification of singularities and topology of Schubert varieties, e.g. the 30-year old problem of the fixed-point-property for Grassmannians. They also propose several additional projects, continuing some of the PI and co-PI's past work in algebraic combinatorics. These projects relate to the recently discovered "cyclic sieving phenomenon", to alternating subgroups of Coxeter groups, and to critical groups of graphs. Some of these projects are outgrowths of the PI's past (NSF-funded) REU's.The appeal of this work is that counting problems, one of the staples of the field of combinatorics, which seem unrelated to algebra or geometry sometimes find the key to their solution and/or application in these other areas. This has been particularly true for the algebraic/geometric objects in this proposal: the Schubert varieties in flag manifolds and Grassmannians. These objects are the key to understanding how lines and planes can tilt and intersect at different angles in the two-dimensional plane, in three-dimensional space, and in higher dimensions.