T-Poisson manifolds and Mirkovic-Vilonen cycles

T-泊松流形和 Mirkovic-Vilonen 循环

• 批准号: 1303124
• 负责人: Allen Knutson
• 金额: $ 18万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303124&HistoricalAwards=false
• 关键词: Poisson; manifolds; Mirkovic; Vilonen; cycles

The PI proposes four projects, the first two closely related. The first is a study of manifolds possessing stratifications with the same good properties possessed by the Bruhat decomposition of a flag manifold. In particular, in this project the PI, with Xuhua He and Jiang-Hua Lu, will explore whether certain other famous stratified spaces (e.g. the 'wonderful compactification' of a Lie group) can similarly be given a stratified atlas of Bruhat cells. The second is about the extent to which the good properties -- Frobenius splitting, Poisson structures, total positivity -- imply one another. The third project (with Joel Kamnitzer) is about giving a cohomological description of the coordinate rings for Mirkovic-Vilonen cycles in loop Grassmannians. If one thinks of representation theory as either about coherent sheaves on flag manifolds (finite-dimensional geometry) or constructible sheaves on loop Grassmannians (infinite-dimensional topology), then this project is looking one level deeper, at coherent sheaves on loop Grassmannians. The fourth project is about extending the 'puzzle' framework of the PI and Tao for Schubert calculus to determine in a positive way the classes of arbitrary positroid subvarieties.Many of the most beautiful spaces considered by mathematicians, and structures on those spaces, have been discovered through symmetry considerations, but perhaps this is akin to looking for one's keys under the lamppost because the light is brightest there. One well-known space and structure is the space of 'flags' of subspaces, with a 'stratification' given by looking at the level of intersection with a fixed flag; any linear transformation preserving the standard flag gives a symmetry of the stratification. My coauthors and I are investigating other spaces in which such symmetries are present only locally, finding unsuspected structure on familiar spaces and stratifications. One of the successes inspiring this study is a stratification of the space of full rank k x n matrices, stratified according to the position of pivots after Gaussian elimination on each rotation of the columns. Without the rotation, this is a 19th century idea. In a separate project, the PI intends to determine properties such as the volume of these strata, extending a century's worth of work on "Schubert calculus", the subcase using only the reflection of the columns.

PI提出了四个项目，前两个项目密切相关。 第一个是对具有相同良好特性的分层的流形的研究。 特别是，在这个项目中，与Xuhua He和Jiang-hua Lu一起，PI将探索是否可以同样地给出某些其他著名的分层空间（例如，谎言组的“奇妙的紧凑”）。 第二个是关于良好特性（Frobenius分裂，泊松结构，完全阳性）的程度。 第三个项目（与Joel Kamnitzer一起）是关于对Loop Grassmannians中Mirkovic-Vilonen循环的坐标环的共同学描述。 如果人们认为代表理论是关于旗杆上的相干滑轮（有限维几何形状）或循环花格拉斯曼尼亚人（无限维拓扑）的可构造滑轮，那么这个项目在一个更深的水平上，在循环grassmannian上的连贯支架上。 第四个项目是为了将PI和TAO的“拼图”框架扩展到Schubert微积分，以积极的方式确定任意的源自源性子变量的类别。数学家所考虑的最美丽的空间，以及这些空间上的结构最美丽的空间，而这些空间是通过对称性考虑发现的，但可能是为了寻找一点点的灯光，因为它是一个亮点，因为它是一个灯光，因为它是一个灯光的灯光。 一个众所周知的空间和结构是子空间的“标志”的空间，通过查看与固定标志的相交水平给出了“分层”。保留标准标志的任何线性变换都会给出分层的对称性。 我和我的合着者正在研究其他空间，在这些空间中，这种对称性仅在本地存在，在熟悉的空间和分层上发现了不受欢迎的结构。 启发这项研究的成功之一是对全等级k x n矩阵的空间进行分层，该空间是根据高斯消除柱旋转后的枢轴的位置进行分层的。 没有轮换，这是一个19世纪的想法。 在一个单独的项目中，PI打算确定诸如这些层的体积之类的属性，从而延长了一个世纪的“舒伯特微积分”的工作价值，这仅使用列的反射。