Topics in number theory, dynamical systems and discrete geometry

最近，数论与其他领域之间发生了许多重要的相互作用，包括动力系统和离散几何。通过填充将数论与离散几何结构相联系导致了与材料科学中的问题的联系。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有相似之处。数论通过模形式与物理理论，如保形场理论有了进一步的联系。数论结构出现在一些物理模型中的相变精确可解模型中。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的研究人员进行富有成效的互动。这笔补助金将用于支持研究生在这些领域的培训。拉加里亚斯博士将研究这些领域之间相互影响的几个独立话题。第一个也是主要的主题继续研究Lerch Zeta函数，它是一个三个变量的函数，在特殊变量上产生Hurwitz Zeta函数和Riemann Zeta函数。将海森堡群及其相关群的表示理论联系起来，旨在将其与各种群上的自同构表示更紧密地联系起来。第二个主题涉及两个探索性项目；一个是关于对称群的一族虚拟表示，它作为多项式(模p)的分裂性质的退化极限而产生，与多项式的判别轨迹的性质有关。它询问是否有一个潜在的几何结构来解释观察到的极限属性，可能与相关的动力系统有关。另一个探索性项目研究多调和模形式，它是具有模不变性的函数，它被拉普拉斯的幂所湮没。第三个主题是研究黎曼曲面上的圆填充，这是离散几何领域的一个主题。它考虑了这类填充在两个方向上的标度极限的公式，即复变量极限和丢番图逼近极限。无论能否准确地做到这一点，提出了一些特殊的问题来初步研究这些领域，包括研究具有有限个圆的刚性填充的曲面。

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.