Schubert and Birkhoff varieties in affine flag varieties

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。Mitchell的项目是从拓扑学、组合学、李理论和代数几何的角度研究仿射旗帜变种。重点研究了Schubert、Birkhoff和Richardson变种，特别是奇异基因座和等变上同调的研究。例如，在早期与Sara Billey的合作中，Mitchell完全确定了仿射Grassmanian中的光滑和合理光滑的Schubert簇；现在的目标是将这些结果推广到更一般的仿射旗簇，并通过确定奇异轨迹来提炼它们。在早期与卢克·古茨威勒的合作中，米切尔完全确定了Birhkhoff品种的同伦类型。然而，关于它们的等变上同调和相关的Richardson簇的许多有趣的问题仍然没有解决，并将在新的研究计划中得到解决。Mitchell的项目位于几何、拓扑学和组合学的交界处。拓扑学是几何学的近亲，但起源更近，它涉及在连续变形下不变的几何对象的性质。近年来，它在物理学、工程学(例如机器人学)、计算机科学和生物学(例如DNA结构)中发现了令人惊讶的应用。组合数学，或“有限数学”，是最直接适用于计算机技术的数学分支。这个项目关注的是构成这些应用程序的基础的纯数学。该项目的关键流行语是“舒伯特多样性”，这个概念不可能用一小段话来解释，但在数学本身已经有了一个半世纪的辉煌历史，并开始在计算机图形学和计算机视觉等领域找到应用。因此，米切尔项目的目标是提高我们对这些引人注目的“舒伯特变数”的几何、拓扑和组合学的理解。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Mitchell's project is an investigation of affine flag varieties from the point of view of topology, combinatorics, Lie theory and algebraic geometry. The focus is on Schubert, Birkhoff and Richardson varieties, especially the study of singular loci and equivariant cohomology. For example, in earlier joint work with Sara Billey, Mitchell completely determined the smooth and rationally smooth Schubert varieties in an affine Grassmannian; now the goal is to extend these results to more general affine flag varieties, and to refine them by determining the singular loci. In earlier joint work with Luke Gutzwiller, Mitchell completely determined the homotopy type of Birhkhoff varieties. However, many interesting problems concerning their equivariant cohomology and associated Richardson varieties remain unsolved, and will be addressed in the new research program.Mitchell's project lies at the interface of geometry, topology and combinatorics. Topology, a cousin of geometry but of much more recent origin, is concerned with properties of geometric objects that are invariant under continuous deformations. In recent years it has found surprising applications to physics, engineering (e.g. robotics), computer science and biology (e.g. structure of DNA). Combinatorics, or ``finite math'', is the branch of mathematics most directly applicable to computer technology. This project is concerned with the pure mathematics that forms the foundation of these applications. The key buzzword attached to the project is ``Schubert variety'', a concept impossible to explain in a short paragraph but having a distinguished history of a century and half within mathematics itself, and with applications beginning to be found in fields such as computer graphics and computer vision. Thus the goal of Mitchell's project is to improve our understanding of the geometry, topology and combinatorics of these remarkable ``Schubert varieties''.