Applied Asymptotic Algebraic Combinatorics

代数组合学利用代数的力量来分析离散结构。它是回答组合问题的一种强有力的方法，经常导致精确而明确的枚举公式，而这些公式不是从第一原理获得的。渐近代数组合学研究的是定义参数极大的离散结构，远远超出日常物理经验的数值范围。在这种情况下，精确的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通常无法获得的有用的近似。渐近组合学的代数方法在大数据时代尤其相关，在这个时代，离散结构在信息领域隐约可见，但处理它们的工具短缺。这项研究的目的是利用代数技术进一步开发渐近组合学中的有用工具。该项目将吸引研究生参与研究。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近分析。在概率方面，PI将以最近的大秩轨积分分析为基础，建立大型随机矩阵的渐近傅立叶分析理论。这项工作的主要目标之一是提供一个工具箱，该工具箱可以统一地应用于大型随机矩阵及其量化的对应物，即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型秩链积分的代数技术来对杨-米尔斯配分函数进行严格的研究，首先重温二维情况，然后通过链环积分转移到更高的维度。这里的一个关键目标是严格理解渐近自由和规范弦对偶的组合学。这个奖项反映了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。

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.