| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 29, 2015 |
| Latest Amendment Date: | June 10, 2017 |
| Award Number: | 1500987 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2015 |
| End Date: | July 31, 2019 (Estimated) |
| Total Intended Award Amount: | $220,000.00 |
| Total Awarded Amount to Date: | $220,000.00 |
| Funds Obligated to Date: |
FY 2016 = $70,000.00 FY 2017 = $80,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
2601 WOLF VILLAGE WAY RALEIGH NC US 27695-0001 (919)515-2444 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
2701 Sullivan Drive; CB 7514 Raleigh NC US 27695-8205 |
| 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): | Combinatorics |
| Primary Program Source: |
01001617DB NSF RESEARCH & RELATED ACTIVIT 01001718DB NSF RESEARCH & RELATED ACTIVIT |
| 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
This research project is in discrete mathematics, namely, the area of mathematics which provides the theoretical underpinnings for computer science as well as more recently for some substantial parts of biology. The PI particularly focuses on developing novel ways of combining geometric and topological techniques and intuition with combinatorial methods. In recent years, the PI has become particularly focused on finding effective ways to study topological-combinatorial structures on spaces of real-valued matrices satisfying naturally arising constraints, for instance matrices in which the determinant as well as all minors are nonnegative. Such spaces arise both in areas of theoretical mathematics such as representation theory and also in applications areas. For instance, they play an important role to our understanding of the relationship between current and voltage in electrical networks. The more theoretical results can sometimes give surprisingly powerful insights into such applications. The project also includes a study of how configurations of distinct points may move around in space without bumping into each other, taking an abstract, representation theoretic perspective. The PI will also continue her work in helping develop the STEM pipeline both through the training of graduate students in combinatorics and also through organizing workshops and other activities to help inspire and foster the development of the next generation of scientists.
The specific projects include: (1) analysis of the homeomorphism type of fibers of maps to totally nonnegative varieties; (2) stability properties for configuration spaces related to the partition lattice via a mixture of poset topology and symmetric function theory; (3) analysis of combinatorial topological structure on spaces of electrical networks; and (4) development of poset-theoretic approaches to polytope diameter bounds for particularly nice classes of polytopes, motivated by complexity questions from operations research regarding linear programming. Many of these projects are collaborative. This work builds upon the PI's past research in topological combinatorics, and particularly in poset topology and in combining ideas of geometric topology with those of combinatorics to study combinatorial topological structure of stratified spaces.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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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.
Intellectual merit:
In her research, the PI studied the structure of naturally arising spaces such as spaces of n by n matrices of real numbers whose determinant is nonnegative and whose minors are also nonnegative, combining combinatorics and topology to get at the structure. Many of these spaces arise as images of functions where one also really wants to understand which inputs lead to a given output, namely to solve the inverse problem. Rather than just having one inverse, there can be an entire space of inverses to a given point, and it can be useful to put structure on this space of inverses. In collaboration with James Davis and Ezra Miller, the PI has conjectured what should be the structure of these spaces of inverses in important cases, and she and her collaborators have proven many pieces of this conjecture.
In other solo work, the PI has devised a new combinatorial perspective to one of the main problems of operations research, namely the problem of better understanding when the simplex method for linear programming will be efficient. The simplex method is used to solve real world problems such as the traveling salesman problem, but there is a large gap between the theoretical results warning how inefficient it can be and the actuality that it is really efficient on many practical problems. The PI observed that one possible explanation may be that certain conditions on the space (called a polytope) where one seeks to optimize a function may force the simplex method to work efficiently on that polytope, even if there exist polytopes where it will be highly non-efficient. The conditions the PI proposes which force the simplex method to be efficient are ones that also make it possible to prove results about the polytope (related to efficiency of the simplex method) by techniques from an entirely different part of discrete mathematics, namely what are known as poset theoretic techniques. The PI proved a number of results in this direction and hopes yet to do more in the future.
The PI also carried out other projects, for instance joint work with Victor Reiner in the area of representation theoretic stability and joint work with Richard Kenyon regarding how to break spaces of electircal networks into pieces that fit together in a nice way combinatorially. She disseminated her results through numerous research talks as well as through publications.
Broader impacts:
The PI has supervised four Ph.D. students under this project, three of whom successfully completed their Ph.D.s in 2019 and one of whom is continuing. She served and continues to serve as a journal editor for Proceedings of the AMS, for Forum of Mathematics Pi, for Forum of Mathematics Sigma and also serves on the advisory board (which is much like an editorial board) for the Springer Graduate Texts in Mathematics book series. She served and continues to serve on the steering committee as well as several local organizing committees for the Triangle Lectures in Combinatorics conference series as well as serving on the organizing committee for an MSRI workshop in Topological Combinatorics. She also has served on and chaired two committee for the American Mathematical Society, namely the AMS Program Committee for the Southeastern Section and the AMS Fellows Selection Committee.
Last Modified: 10/27/2019
Modified by: Patricia L Hersh
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