Schubert varieties: Combinatorics, Computation and Geometry

提议者将在一套主要项目上进行合作，这些项目涉及组合学、几何学和Grassmannian拓扑学、旗流形及其舒伯特变种。这些变种是跨越许多经典数学领域的核心问题，如代数拓扑学，枚举几何和表示论。他们希望解决一些棘手的问题，如奇点的分类和舒伯特簇的拓扑，例如30年前的Grassmannian簇的不动点性质问题。他们还提出了几个额外的项目，继续PI和co-PI过去在代数组合学方面的工作。 这些项目涉及到最近发现的“循环筛选现象”，交替的考克斯特群的子群，并临界群的图形。 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，组合数学领域的斯台普斯之一，似乎与代数或几何无关，有时会找到解决方案的关键和/或在这些其他领域的应用。这对于这个提议中的代数/几何对象尤其如此：旗流形中的舒伯特簇和格拉斯曼簇。 这些物体是理解线和平面如何在二维平面、三维空间和更高维度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.