Asymptotic Representation Theory, Enumeration, and Combinatorial Probability

该项目涉及纯数学几个领域的问题，如数论、整数及其性质的研究；表征理论，对称的研究；以及概率论，对随机性的数学研究。这些问题似乎彼此之间没有什么共同之处，它们似乎也没有任何明显的应用，然而，深入研究就会发现它们之间有许多微妙的联系，并可能应用于应用科学和理论科学的其他领域。例如，数学家在20世纪80年代就逻辑学家查尔斯·道奇森（Charles doddgson）（又名刘易斯·卡罗尔（Lewis Carroll））在19世纪提出的一个模糊算法提出了“有多少个N阶的交替符号矩阵？”这个问题，结果证明与“方冰”有关。方冰是几年前由英国研究人员通过实验发现的一种奇特的水冰形式。同样的，Witten zeta函数，这个研究者计划研究的与某些对称类型的枚举相关的函数，之前被其他研究者（包括物理学家Edward Witten）在与量子场论问题相关的不同背景下研究过。正是这些联系，研究者希望通过他的研究来阐明，这使得纯数学成为人类知识进步的一个富有成果的研究领域。研究者建议研究的具体问题是在枚举、代数和渐近组合学的界面上的问题，与数学的其他分支有联系，特别是表示论、概率论、渐近分析和数论。例如，一个问题涉及到渐近表示理论的一个新方向，即当n变大时，求李群的n维表示数的渐近公式。研究者最近的论文解决了群SU(3)的这个问题，使用了一个与群相关的所谓的Witten zeta函数的困难分析，突出了这个问题与解析数论和模形式理论中看似无关的问题之间的一些有趣的联系。在该项目的另一部分，研究者建议撰写一本关于交替符号矩阵的专著，交替符号矩阵自20世纪80年代被发现以来一直是人们非常感兴趣的主题，并且与组合学，概率论和统计物理中的许多当代研究主题相关。该奖项反映了美国国家科学基金会的法定使命，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。

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.