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.
