| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 2, 2018 |
| Latest Amendment Date: | July 18, 2023 |
| Award Number: | 1808376 |
| Award Instrument: | Standard 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, 2024 (Estimated) |
| Total Intended Award Amount: | $150,000.00 |
| Total Awarded Amount to Date: | $163,776.00 |
| Funds Obligated to Date: |
FY 2020 = $6,888.00 FY 2021 = $6,888.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
100 INSTITUTE RD WORCESTER MA US 01609-2280 (508)831-5000 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
100 Institute Road Worcester MA US 01609-2280 |
| 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: |
01001819DB NSF RESEARCH & RELATED ACTIVIT 01002021DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This project investigates combinatorial objects fundamental to various areas of communications, information theory, networks, and numerous topics in pure mathematics. The team, consisting of the Principal Investigator (PI) and a PhD student will study association schemes and configurations in real and complex space. The project is motivated by applications in the theory of error-correcting codes, cryptography, quantum information theory, knot theory, extremal networks, finite geometry and spherical and projective codes. Association schemes and related tools play a fundamental role in these and other areas, for example guiding us to the best known efficiency bounds for binary error-correcting block codes used in most digital communications devices. As digital technologies grow in scale and complexity, we see increasing need for algebraic tools of this sort that extract important structural information efficiently from data. The project's broader impacts include training of highly qualified personnel and outreach to students and teachers in local schools.
Progress on association schemes over the past 50 years has been phenomenal, and the variety of new applications that the theory handles has increased with each passing decade. Yet some fundamental problems, regarding cometric association schemes for example, remain unresolved with few new ideas emerging until recently. This project explores powerful mathematical tools, ranging from algebraic geometry to algebraic topology, to forge a stronger and more versatile theory of association schemes that will be equipped to attack existing open questions - both theoretical and applied - as well as future challenges that are likely to be framed in this general combinatorial language. Specific problems to be attacked include: efficient description of the ideal of polynomials vanishing on the set of columns of the first primitive idempotent of a cometric scheme; constructions (via finite geometry and design theory) and bounds for systems of lines with few angles; bounds on efficiency parameters of association schemes and codes; determining the structure of additive completely regular error-correcting codes. The mathematical tools to be employed have mainly been developed by the algebraic combinatorics community, including linear-algebraic and ring-theoretic techniques, as well as zero-dimensional ideals in polynomial rings and discrete homotopy.
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.
This project explores pure mathematics and certain applications of that theory. The main topic of study is the theory of association schemes which arose out of the design of statistical experiments, the study of finite groups and finite geometry, and the theory of error-correcting codes. The intellectual merit includes advances in the application of elementary algebraic geometry to spherical codes and designs, a new study of quantum state transfer on graphs, tools for the application of Delsarte theory to codes and designs in association schemes with irrational eigenvalues, and the development of a new algebraic tool called a ``scaffold'' which allows for simplified computations of tensors related to association schemes and similar structures. The scaffold formalism also illuminates possible avenues for extending the theory. An important special case of the PI's conjecture on scaffold duality was soon verified by Liang, et al. The versatility of the theory is demonstrated in the PI's publication with Chan that identifies exponentially large families of non-isomorphic graphs that are yet quantum isomorphic, shedding light on the power of quantum entanglement through these quantum games.
The broader impacts of the work include potential applications to communications, cryptology, and quantum information theory as well as the training of highly qualified personnel. The grant funded one PhD student, one postdoctoral fellow (for summer research), and three undergraduate summer researchers. But the project also facilitated the mentorship of another postdoctoral researcher, a visiting PhD student, an internally funded summer resaerch student, eight undergraduate senior thesis students, and three students informally advised on an unfunded summer research team. Of these, at least three are now employed in government positions related to national security and several more are pursuing graduate degrees.
Last Modified: 04/02/2025
Modified by: William J Martin
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