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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。该项目解决了两个重要而及时的问题。首先，本工作考虑如何确保云存储和计算的可靠性、效率和安全性。纠错码是一种数学方法，它将冗余添加到信息中，以便在错误和擦除不断发生的情况下，仍能正确存储和通信数据。PI将使用数论和几何来扩展局部可恢复码的理论并创建示例，局部可恢复码是一种受云计算设施的特殊挑战推动的纠错码。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职业生涯早期的数学教师建立协作孵化器，开发基于探究的学习模块和附带的说明性文章，以扩大本科离散数学教育的范围并拓宽进入数学的大门，开发并使用这些材料向正在过渡到大学的早期本科生(包括许多来自代表不足的群体和低收入家庭的学生)开发和教授一门课程，以及指导本科生进行暑期研究和高级论文项目。更具体地说，PI计划利用有限域上曲线和高维变化的结构来创建具有可用性的局部可恢复的代数几何代码，即评估代码，其中任何代码字中的每个位置都可以以多种方式有效恢复。此外，该项目试图扩展和统一局部可恢复码理论，寻求在私人信息检索中的应用，并解决算术几何中的相关问题。在应用领域，PI建议解决与科罗拉多州当前社会问题相关的问题，改进和完善加权图形技术，为集合分析创建更具代表性的随机地图，扩大集合分析解决的问题的范围，并解决图论中的潜在问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.