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Asymptotic Representation Theory, Enumeration, and Combinatorial Probability

渐近表示理论、枚举和组合概率

基本信息

批准号：
1800725
负责人：
Dan Romik
金额：
$ 20万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800725&HistoricalAwards=false
关键词：
Asymptotic Representation Theory Enumeration Combinatorial

项目摘要

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.

该项目涉及的问题从几个领域的纯数学，如数论，研究的整数及其性质;表示论，研究对称性;和概率论，数学研究的随机性。这些问题似乎彼此之间没有什么共同之处，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，以及它们在应用科学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？数学家们在20世纪80年代提出了一个问题，与19世纪逻辑学家查尔斯·道奇森（又名刘易斯·卡罗尔）提出的一个模糊算法有关，结果发现它与“方形冰”有关，方形冰是一种奇异的水冰形式，几年前由英国研究人员通过实验发现。类似地，研究者计划研究的维滕zeta函数与某些类型的对称性的计数有关，其他研究者（包括物理学家爱德华维滕）以前在与量子场论问题有关的不同背景下研究过。正是这些类型的联系，调查员希望通过他的研究阐明，并使纯数学这样一个富有成果的研究领域，为人类知识的进步。具体问题的调查员提出研究的接口问题的枚举，代数和渐近组合，与其他分支的数学，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论的一个新方向，即当n变大时，寻找李群的n维表示数的渐近公式。研究者最近的论文解决了SU（3）群的这个问题，使用了与群相关的所谓维滕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，研究人员建议作者的专著交替符号矩阵，这一直是极大的兴趣，因为他们在20世纪80年代发现的主题，并与许多当代研究课题组合，该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的知识支持。优点和更广泛的影响审查标准。