Topics in number theory, dynamical systems and discrete geometry

最近，数论与其他领域（包括动力系统和离散几何）之间出现了许多重要的相互作用。数论通过堆积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有相似之处。 数论与物理理论（例如通过模形式的共形场论）有进一步的联系。数论结构出现在物理模型中一些可精确求解的相变模型中。该提案调查了多个领域之间可能接触的领域，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。这笔赠款将支持这些领域研究生的培训。 Lagarias 博士将研究这些领域之间相互作用的几个独立主题。第一个也是主要主题继续研究 Lerch zeta 函数，它是三个变量的函数，在专门变量上产生 Hurwitz zeta 函数和 Riemann zeta 函数。与海森堡群和相关群的表示理论建立了联系，该项目旨在将其与各种群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；其中一个涉及对称群的虚拟表示的某个族，它作为多项式分裂属性（模 p）的简并极限而出现，与多项式判别轨迹的属性相关。它询问是否存在解释观察到的极限属性的潜在几何结构，可能具有相关的动力系统。另一个探索性项目研究多调和模形式，它们是具有模不变性的函数，被拉普拉斯算子的幂所消除。第三个主题是研究黎曼曲面上的圆堆积，这是离散几何领域的一个主题。它考虑了此类填料在两个方向上的缩放极限的制定，即复杂变量极限和丢番图近似极限。无论这是否可以精确地完成，都提出了特殊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.