LEAPS-MPS: Foundational Connections in Number Theory, Coding Theory, Graphs, and Their Applications

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project addresses two important and timely problems. First, this work considers how to ensure the reliability, efficiency, and security of cloud storage and computing. Error-correcting codes are mathematical methods of adding redundancy to information so that the data is stored and communicated correctly, despite the fact that errors and erasures occur constantly. The PI will use number theory and geometry to expand the theory of and create examples of locally recoverable codes, a type of error-correcting code motivated by the special challenges of cloud computing facilities. Secondly, this project will use the mathematical framework of ensemble analysis in new settings and applications, including applications with a societal impact. Activities to broaden participation include a collaboration incubator for new and early-career mathematics faculty members, developing inquiry-based learning modules and accompanying expository articles to broaden the scope of undergraduate discrete math education and widen the doorway into mathematics, developing and teaching a course using these materials to early undergraduates who are transitioning to college including many students from underrepresented groups and low-income families, and mentoring undergraduates in summer research and senior thesis projects.More specifically, the PI plans to exploit the structure of curves and higher-dimensional varieties over finite fields to create locally recoverable algebraic geometry codes with availability, i.e. evaluation codes where each position in any codeword is efficiently recoverable in many ways. Further, this project seeks to expand and unite the theory of locally recoverable codes, to pursue applications in private information retrieval, and to address relevant questions in arithmetic geometry. In the application arena, the PI proposes to address questions that are relevant to current societal issues in Colorado, to improve and refine weighted graph techniques that create more representative random maps for ensemble analysis, to broaden the scope of questions addressed by ensemble analysis, and to solve underlying problems in graph theory.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.