Topics in number theory, dynamical systems and discrete geometry

近年来，数论与其他领域，包括动力系统和离散几何之间有许多重要的相互作用。数论通过填料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ζ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共形场论有进一步的联系。数论结构出现在物理模型中一些完全可解的相变模型中。该提案调查了几个领域之间可能的接触领域，这可能导致与物理学和材料科学研究人员的富有成效的互动。该补助金将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，主题继续研究lach zeta函数，它是一个三个变量的函数，在专门化变量上产生Hurwitz zeta函数和Riemann zeta函数。与海森堡群和相关群的表征理论建立了联系，该项目旨在将其与各种群上的自同构表征更紧密地联系起来。第二个主题涉及两个探索性项目；一类是关于对称群的虚表示，它是多项式（模p）的分裂性质的退化极限，与多项式的判别轨迹的性质有关。它询问是否存在一个潜在的几何结构来解释观察到的极限性质，可能还有一个相关的动力系统。另一个探索性项目研究多谐模形式，它是具有模不变性的函数，被拉普拉斯函数的幂湮灭。第三个课题是研究黎曼曲面上的圆填充，这是一个离散几何领域的课题。考虑了这类填料在两个方向上的标度极限的表述，即复变量极限和丢番图近似极限。无论是否可以精确地做到这一点，提出了一些特殊的问题来初步研究这些领域，包括研究具有有限数量圆的刚性填料的表面。

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.