| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 1, 2018 |
| Latest Amendment Date: | August 12, 2020 |
| Award Number: | 1764012 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Stefaan De Winter
sgdewint@nsf.gov (703)292-2599 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2018 |
| End Date: | July 31, 2023 (Estimated) |
| Total Intended Award Amount: | $187,269.00 |
| Total Awarded Amount to Date: | $187,269.00 |
| Funds Obligated to Date: |
FY 2020 = $56,246.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
4333 BROOKLYN AVE NE SEATTLE WA US 98195-1016 (206)543-4043 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
WA US 98195-0001 |
| 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: |
01002021DB 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
Combinatorics is an area of mathematics which is quintessential for applications in computer science, biology, physics, chemistry, and industry. Optimized algorithms based on combinatorics have revolutionized business decisions in our lifetime. Current research is uncovering connections to all areas of mathematics and science. This project focuses specifically on interdisciplinary applications of combinatorics in connection with problems in algebra,geometry, probability, and computer science. Combinatorial connections have been at the core of the investigator's prior work and continue to inspire innovation and collaboration.
This project describes three main themes for research. The first relates to the classical study of the coinvariant algebra and its representation theory. The problem is to study the asymptotics of the underlying decomposition into irreducible symmetric group modules. The methods of attack include tools from combinatorics, algebra and probability theory. The second studies a newly proposed family of symmetric functions related to the matroid of 0-1 vectors in n-dimensional space. Conjectures and theorems in this direction use tools from algebraic geometry. This research has connections to physics and economics. The third topic, which is a mixture of combinatorics, theoretical computer science and discrete geometry, pertains to placements of circles in the plane with a wrapping condition. This area of research was inspired by the general discrete geometry problem of finding an appropriate polygonization of a region in the plane which appears in graphics and optimization.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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.
The PI and collaborators have extensively studied asymptoticdistributions related to the descent statistic for standard Youngtableux, semistandard tableaux, and forests. This direction lead usto define and develop the cyclotomic generating functions or CGFs.The CGF's include many famous polynomials in the liturature includingHilbert series of Cohen-Macalay rings, q-analogs of alternating signmatrices, and the length generating functions of Weyl groups.
The PI has been conduting research on positroid varieties with JordanWeaver. Positroids are certain representable matroids originallystudied by Postnikov in connection with the totally nonnegativeGrassmannian and now used widely in algebraic combinatorics. Thepositroids give rise to determinantal equations defining positroidvarieties as subvarieties of the Grassmannian variety. Rietsch,Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomologicalproperties of these varieties. In this direction, we studiedthe geometric properties of positroid varieties by establishingseveral equivalent conditions characterizing smooth positroidvarieties using a variation of pattern avoidance defined on decoratedpermutations, which are in bijection with positroids. This allows usto give two formulas for counting the number of smooth positroidsalong with two $q$-analogs. Furthermore, we give a combinatorialmethod for determining the dimension of the tangent space of apositroid variety at key points using an induced subgraph of theJohnson graph. We also give a Bruhat interval characterization ofpositroids. An extended abstract on this work was accepted to FPSAC2022 in India as a talk, which is a high honor in our field.
Eight papers have been published and two more submitted. Eightgraduate students and 12 undergraduates have been mentored by the PIin research in mathematics. The PI works to increase diversity in themathematical workforce through mentoring.
Last Modified: 01/24/2024
Modified by: Sara C Billey
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