Affine flags, folded galleries and euclidean reflection groups

这个项目的目的是回答重要的几何问题的子品种的仿射旗品种是根本的代数几何和数论。 它将回答这些基本问题的中心对象的数学，仿射旗及其子空间，使用强大的新方法，结合联合收割机的想法，从几何和代数。 该项目预计将包括找到这些子品种的非空模式和它们的尺寸公式。 它将发展和应用新的方法，结合联合收割机折叠画廊和几何的欧几里德反射群，这些方法将有应用在代数组合学和表示论。 该项目还将激发几何群论，一个新的和快速增长的领域，和代数几何，代数组合学和表示论的经典领域之间的生产联系。

This project aims to answer important geometric questions about subvarieties of the affine flag variety which are fundamental to algebraic geometry and number theory. It will answer basic questions about these central objects of mathematics, affine flags and their subspaces, using powerful new methods which combine ideas from geometry and algebra. The project expects to include finding the patterns of non-emptiness of these subvarieties and formulae for their dimension. It will develop and apply new methods which combine folded galleries and the geometry of Euclidean reflection groups, and these methods will have applications in algebraic combinatorics and representation theory. The project will also inspire productive connections between geometric group theory, a new and fast-growing area, and the classical fields of algebraic geometry, algebraic combinatorics and representation theory.