Affine flags, folded galleries and euclidean reflection groups

这个项目的目的是回答有关仿射旗簇的子簇的重要几何问题，这些子簇是代数几何和数论的基础。它将使用结合了几何和代数思想的强大的新方法，回答关于这些数学中心对象、仿射旗帜及其子空间的基本问题。该项目预计将包括寻找这些亚种的非空性模式和它们的维度公式。它将开发和应用将折叠画廊和欧几里德反射群几何相结合的新方法，这些方法将在代数组合学和表示论中得到应用。该项目还将促进几何群论这一新的快速增长的领域与代数几何、代数组合学和表示论等经典领域之间的富有成效的联系。

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.