T-Poisson manifolds and Mirkovic-Vilonen cycles

PI提出了四个项目，前两个密切相关。第一个是研究具有与标志流形的Bruhat分解具有相同优良性质的分层的流形。特别是，在这个项目中，PI将与何旭华和陆江华一起探索是否某些其他著名的分层空间（例如李群的“奇妙紧化”）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——弗罗贝尼乌斯分裂、泊松结构、总正性——相互暗示的程度。第三个项目（与Joel Kamnitzer合作）是关于给出环格拉斯曼中Mirkovic-Vilonen环的坐标环的上同调描述。如果一个人认为表征理论要么是关于旗形流形上的相干束（有限维几何），要么是关于格拉斯曼环上的可构造束（无限维拓扑），那么这个项目就在更深入地研究格拉斯曼环上的相干束。第四个项目是将PI和Tao的“谜题”框架扩展到舒伯特微积分中，以一种积极的方式确定任意正类子变种的类别。数学家们所考虑的许多最美丽的空间，以及这些空间上的结构，都是通过对称的考虑而发现的，但也许这就像在灯柱下寻找钥匙一样，因为那里的光线最亮。一个众所周知的空间和结构是子空间的“旗帜”空间，通过观察与固定旗帜相交的水平来给出“分层”；任何保持标准标志的线性变换都给出了分层的对称性。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意想不到的结构。启发这项研究的成功之一是对全秩k x n矩阵的空间进行分层，在每次旋转列时，根据高斯消去后的枢轴位置分层。没有旋转，这是19世纪的想法。在另一个单独的项目中，PI打算确定这些地层的体积等属性，扩展一个世纪以来对“舒伯特演算”的研究，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.