Applied Asymptotic Algebraic Combinatorics

应用渐近代数组合学

• 批准号: 2054488
• 负责人: Jonathan Novak
• 金额: $ 24万
• 依托单位: University of California-San Diego
• 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: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054488&HistoricalAwards=false
• 关键词: Applied; Asymptotic; Algebraic; Combinatorics

Algebraic combinatorics leverages the power of algebra to analyze discrete structures. It is a powerful approach to answering combinatorial questions, often leading to exact and explicit enumerative formulas that are not obtainable from first principles. Asymptotic algebraic combinatorics deals with discrete structures whose defining parameters are extremely large, far exceeding the numerical range of everyday physical experience. In this setting, exact formulas become unwieldy and unusable; asymptotic algebraic combinatorics leverages algebraic methods to obtain useful approximations that are typically not accessible. The algebraic approach to asymptotic combinatorics is especially pertinent in the age of big data, where discrete structures loom large over the information landscape, but the tools to handle them are in short supply. This research project intends employ algebraic techniques to further develop useful tools in asymptotic combinatorics. The project will involve graduate students in the research.This project aims to develop new algebraic methods for the asymptotic analysis of large structures that appear in probability and mathematical physics. On the probabilistic side, the PI will build a theory of asymptotic Fourier analysis for large random matrices using recent analysis of large rank orbital integrals as a foundation. One of the main goals of this endeavor is to provide a toolbox that can be uniformly applied to both large random matrices and their quantized counterparts, random representations of large Lie groups. On the mathematical physics side, the PI plans to use the algebraic techniques underlying recent analysis of large rank link integrals to undertake a rigorous study of Yang-Mills partition functions, first revisiting the two-dimensional case and then moving to higher dimensions via link integrals. A key goal here is to rigorously understand combinatorics of asymptotic freedom and gauge-string dualities.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.

代数组合学利用代数的力量来分析离散结构。这是一个强大的方法来回答组合问题，往往导致准确和明确的枚举公式，是无法获得的第一原则。渐近代数组合学处理离散结构，其定义参数非常大，远远超过日常物理经验的数值范围。在这种情况下，精确公式变得笨拙和不可用;渐近代数组合学利用代数方法来获得有用的近似值，这些近似值通常无法访问。渐近组合学的代数方法在大数据时代尤其重要，在这个时代，离散结构在信息领域中显得很重要，但处理它们的工具却很短缺。本研究计画拟利用代数技巧，进一步发展渐近组合学的有用工具。该项目将邀请研究生参与研究。该项目旨在开发新的代数方法，用于概率和数学物理中出现的大型结构的渐进分析。在概率方面，PI将使用最近的大秩轨道积分分析作为基础，建立大型随机矩阵的渐近傅立叶分析理论。这项奋进的主要目标之一是提供一个工具箱，可以统一应用于大型随机矩阵及其量化对应物，大型李群的随机表示。在数学物理方面，PI计划使用最近分析大秩链接积分的代数技术来进行杨-米尔斯配分函数的严格研究，首先重新审视二维情况，然后通过链接积分转向更高维度。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。