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

LEAPS-MPS：数论、编码理论、图及其应用的基础联系

基本信息

批准号：
2137661
负责人：
Beth Malmskog
金额：
$ 24.99万
依托单位：
Colorado College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-15 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137661&HistoricalAwards=false
关键词：
LEAPS MPS Foundational Connections Number

项目摘要

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.

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。该项目解决了两个重要而及时的问题。首先，这项工作考虑如何确保云存储和计算的可靠性，效率和安全性。纠错码是一种为信息增加冗余的数学方法，以便在错误和擦除不断发生的情况下正确存储和传输数据。PI将使用数论和几何来扩展本地可恢复代码的理论并创建本地可恢复代码的示例，这是一种基于云计算设施的特殊挑战的纠错代码。其次，该项目将在新的环境和应用中使用集合分析的数学框架，包括具有社会影响的应用。扩大参与的活动包括为新的和早期职业数学教师建立一个合作孵化器，开发基于探究的学习模块和附带的简要文章，以扩大本科离散数学教育的范围，拓宽数学的大门，开发和教学课程使用这些材料的早期本科生谁是过渡到大学，包括许多学生从代表性不足的群体和低，更具体地说，PI计划利用有限域上的曲线结构和高维变种来创建具有可用性的局部可恢复的代数几何代码，即任何码字中的每个位置都可以以多种方式有效地恢复的评估代码。此外，该项目旨在扩大和统一的本地可恢复代码的理论，追求在私人信息检索的应用，并解决算术几何的相关问题。在应用竞技场中，PI建议解决与科罗拉多当前社会问题相关的问题，改进和完善加权图技术，为集成分析创建更具代表性的随机地图，扩大集成分析解决的问题范围，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。