| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 3, 2013 |
| Latest Amendment Date: | September 3, 2013 |
| Award Number: | 1303124 |
| Award Instrument: | Standard Grant |
| Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2013 |
| End Date: | August 31, 2016 (Estimated) |
| Total Intended Award Amount: | $180,000.00 |
| Total Awarded Amount to Date: | $180,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
341 PINE TREE RD ITHACA NY US 14850-2820 (607)255-5014 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
515 Malott Hall Ithaca NY US 14853-4201 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
ALGEBRA,NUMBER THEORY,AND COM, Combinatorics |
| Primary Program Source: |
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| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
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.
PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
In his thesis 11 years ago, my student Joel Kamnitzer described a family of higher-dimensional polyhedra in very down-to-earth terms -- specifying the angles of the edges, and constraints on the edge lengths. This was in contrast to the lofty origins of these "MV polyhedra", as shadows (in a rather literal sense) of much larger-dimensional algebro-geometric loci, "MV cycles", most naturally found inside an infinite-dimensional space. As so often in mathematics, these shadows bear less information but are easier to compute.
Several years later, he found a completely unrelated-looking construction of the same polyhedra, as the convex hulls of certain integer vectors (lists of certain dimensions). Our work together has been about souping this second construction up from the MV polyhedra to the actual MV cycles.
Much of our progress has come through recent connections of physicists' "mirror symmetry" in 3 dimensions to corresponding dualities in mathematics.
My other project concerns "atlases on stratified spaces". An "atlas", in this context, is a way to locally understand a more complicated space, say a sphere, by covering it with simple spaces, say two copies of the plane (from stereographic projection from the north and south poles). So all the difficulty (e.g. the nonorientability of the Möbius strip) is due to global issues about how the covering is glued together, not local ones.
A "stratified space" is one with a list of progressively smaller subspaces obeying more stringent conditions, such as the sphere, the equator, and a point on the equator. A more pertinent example is the 4-d family of all lines in xyz-space, containing those lines in the xy-plane, or those lines through the origin, or those lines obeying both conditions.
There is a particularly friendly family of stratified spaces, called "Bruhat cells", generalizing this example just described to higher dimensions; these are the building blocks that we use (in analogy to the two copies of the plane used in the sphere's atlas). Jiang-Hua Lu, Xuhua He, and I showed that various famous stratified spaces (such as the quite aptly named "wonderful compactification of a group") can be given atlases made out of these Bruhat cells, such that the correspondences (analogous to the stereographic projections) respect the stratifications. So once again, the local behavior is under control, and the interest moves to the global behavior or equivalently to study of how the covering is glued together.
Last Modified: 12/14/2016
Modified by: Allen Knutson
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