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Topics in number theory, dynamical systems and discrete geometry

数论、动力系统和离散几何主题

基本信息

批准号：
1401224
负责人：
Jeffrey Lagarias
金额：
$ 32.19万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2017-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401224&HistoricalAwards=false
关键词：
Topics number theory dynamical systems

项目摘要

There have recently been many important interactions between number theory and other fields, including dynamical systems and discrete geometry. The connection of number theory with discrete geometry structures via packings leads to connections with problems in materials science. Problems in number theory involving zeta functions and distribution of primes have parallels with problems in mathematical physics. There are further connections of number theory with physical theories such as conformal field theory through modular forms. Number theory structure appears in some exactly solvable models of phase transitions in physical models. The proposal investigates several possible areas of contact between fields, which may lead to fruitful interactions with researchers in physics and material sciences. The grant will support the training of graduate students in these areas. Dr. Lagarias will investigate several independent topics interacting among these fields. The first, and main, topic continues the investigation of the Lerch zeta function, which is a function of three variables, that on specializing variables yields the Hurwitz zeta function and Riemann zeta function. A connection is made with the representation theory of the Heisenberg group and related groups, and the project aims to connect it more closely to automorphic representations on various groups. A second topic concerns two exploratory projects; one concerns a certain family of virtual representations of the symmetric groups, which arises as a degenerate limit of splitting properties of polynomials (modulo p), associated to properties of the discriminant locus of the polynomial. It asks if there is an underlying geometric structure explaining the observed limit properties, possibly with an associated dynamical system. Another exploratory project studies polyharmonic modular forms, which are functions with modular invariance that are annihilated by a power of the Laplacian. A third topic is to investigate of circle packings on Riemann surfaces, a topic in the area of discrete geometry. It considers formulation of scaling limits of such packings in two directions, a complex variables limit and a Diophantine approximation limit. Whether or not this can be done precisely, special questions are proposed to initially investigate these areas, including study of surfaces having rigid packings with a finite number of circles.

最近有许多重要的相互作用之间的数论和其他领域，包括动力系统和离散几何。数论通过填充与离散几何结构的联系导致了与材料科学问题的联系。数论中涉及zeta函数和素数分布的问题与数学物理中的问题相似。数论与物理理论有进一步的联系，例如通过模形式的共形场论。在物理模型中，一些相变的精确可解模型出现了数论结构。该提案调查了几个可能的领域之间的接触，这可能会导致与物理学和材料科学研究人员的富有成效的互动。这笔赠款将用于支持这些领域的研究生培训。Lagarias博士将研究这些领域之间相互作用的几个独立主题。第一个，也是主要的，主题继续调查的Lerch zeta函数，这是一个函数的三个变量，这对专业化变量产生的Hurwitz zeta函数和黎曼zeta函数。与海森堡群和相关群的表示理论建立了联系，该项目旨在将其与各种群的自守表示更紧密地联系起来。第二个主题涉及两个探索性的项目;一个涉及一定的家庭的虚拟表示的对称群，这是作为一个退化的限制分裂性能的多项式（模p），相关的属性的判别轨迹的多项式。它询问是否有一个潜在的几何结构解释所观察到的极限性质，可能与相关的动力系统。另一个探索性项目研究多重调和模形式，它是具有模不变性的函数，这些模不变性被拉普拉斯算子的幂湮没。一第三个主题是调查黎曼曲面上的圆填充，离散几何领域的一个主题。它认为制定的标度限制，这样的包装在两个方向，一个复杂的变量限制和丢番图近似限制。无论是否可以做到这一点，精确地说，提出了一些特殊的问题，初步调查这些领域，包括研究表面具有刚性填料与有限数量的圆。