T-Poisson manifolds and Mirkovic-Vilonen cycles

PI提出了四个项目，前两个密切相关。 第一个是研究的流形拥有分层具有相同的良好性质所拥有的Bruhat分解的旗流形。 特别是，在这个项目中，PI与Xuhua He和Jiang-Hua Lu将探索某些其他著名的分层空间（例如李群的“奇妙紧化”）是否可以类似地给出Bruhat细胞的分层图谱。 第二个是关于好的性质-弗罗贝纽斯分裂，泊松结构，总积极性-相互暗示的程度。 第三个项目（与Joel Kamnitzer合作）是关于给出循环格拉斯曼圈中Mirkovic-Vilonen圈的坐标环的上同调描述。 如果人们认为表示论要么是关于旗形流形（有限维几何）上的相干层，要么是关于循环格拉斯曼（无限维拓扑）上的可构造层，那么这个项目正在更深一层地研究循环格拉斯曼上的相干层。 第四个项目是关于扩展舒伯特演算的PI和Tao的“谜题”框架，以确定任意正向子簇的类。数学家认为的许多最美丽的空间以及这些空间上的结构都是通过对称性的考虑发现的，但这可能类似于在灯柱下寻找钥匙，因为那里的光线最亮。 一个著名的空间和结构是子空间的“标志”空间，通过查看与固定标志相交的水平来给出“分层”;任何保持标准标志的线性变换都会给出分层的对称性。 我和我的合著者正在研究其他空间，在这些空间中，这种对称性只在局部存在，在熟悉的空间和分层上发现了意料之外的结构。 激励这项研究的成功之一是一个分层的空间满秩k × n矩阵，分层后的高斯消除列的每次旋转的枢轴的位置。 如果没有旋转，这是一个世纪的想法。 在一个单独的项目中，PI打算确定这些地层的体积等属性，延长了世纪的“舒伯特微积分”工作的价值，该子案例仅使用柱的反射。

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.