Asymptotic Representation Theory, Enumeration, and Combinatorial Probability

该项目涉及纯数学多个领域的问题，例如数论、整数及其性质的研究；表示论，对称性研究；和概率论，随机性的数学研究。这些问题似乎彼此之间没有什么共同点，似乎也没有任何明显的应用，但更深入的研究揭示了它们之间的许多微妙联系以及对应用和理论科学其他领域的可能应用。例如，问题“存在多少个 N 阶交替符号矩阵？” 20 世纪 80 年代，数学家提出了一个与逻辑学家查尔斯·道奇森（又名刘易斯·卡罗尔）在 19 世纪提出的晦涩算法有关的问题，结果证明该算法与“方冰”有关，“方冰”是一种奇特的水冰形式，几年前由英国研究人员通过实验发现。同样，研究人员计划结合某些类型的对称性的枚举来研究 Witten zeta 函数，其他研究人员（包括物理学家 Edward Witten）之前曾在与量子场论问题相关的不同背景下研究过该函数。研究者希望通过他的研究阐明这些联系，并使纯数学成为人类知识进步的富有成果的研究领域。研究者建议研究的具体问题是枚举、代数和渐近组合学的接口问题，与数学的其他分支，特别是表示论、概率论、渐近分析和数论有联系。例如，一个问题涉及渐近表示理论的一个新方向，即随着 n 变大，找到李群的 n 维表示数量的渐近公式。研究人员最近的论文通过对与该群相关的所谓 Witten zeta 函数进行困难分析，解决了 SU(3) 群的这个问题，强调了该问题与解析数论和模形式理论中看似不相关的问题之间的一些有趣的联系。在该项目的另一部分中，研究人员建议撰写一本关于交替符号矩阵的专着，自从 20 世纪 80 年代发现交替符号矩阵以来，该矩阵一直是人们极大兴趣的主题，并且与组合数学、概率论和统计物理学中的许多当代研究主题相关。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project concerns problems from several areas of pure mathematics such as number theory, the study of the integers and their properties; representation theory, the study of symmetry; and probability theory, the mathematical study of randomness. These problems appear to have little in common with each other, nor do they seem to have any obvious applications, and yet a deeper look reveals many subtle connections between them and possible applications to other areas of applied and theoretical science. For example, the question "how many alternating sign matrices of order N exist?" which was asked by mathematicians in the 1980s in connection with an obscure algorithm proposed by the logician Charles Dodgson (aka Lewis Carroll) in the 19th century turned out to have connections to "square ice," an exotic form of water ice that was detected experimentally a few years ago by British researchers. Similarly, the Witten zeta function, which the investigator plans to study in connection with the enumeration of certain types of symmetry, was previously studied by other researchers (including the physicist Edward Witten) in a different context related to problems in quantum field theory. It is these sorts of connections that the investigator hopes to shed light on through his research, and which make pure mathematics such a fruitful area of study for the advancement of human knowledge.The specific problems the investigator proposes to study are questions at the interface of enumerative, algebraic and asymptotic combinatorics, with connections to other branches of mathematics, notably representation theory, probability theory, asymptotic analysis, and number theory. For example, one problem involves a new direction in asymptotic representation theory, namely that of finding asymptotic formulas for the number of n-dimensional representations of a Lie group as n grows large. The investigator's recent paper solved this problem for the group SU(3) using a difficult analysis of the so-called Witten zeta function associated with the group, which highlighted some intriguing connections between the problem and seemingly unrelated questions in analytic number theory and the theory of modular forms. In an additional part of the project, the investigator proposes to author a monograph on alternating sign matrices, which have been the subject of great interest since their discovery in the 1980s and are related to many contemporary research topics in combinatorics, probability theory and statistical physics.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.