Topics in number theory, dynamical systems and discrete geometry

数论、动力系统和离散几何主题

• 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; Topics; number; theory; dynamical; systems

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，数字理论与其他领域之间存在许多重要的相互作用，包括动态系统和离散的几何形状。数字理论与通过包装的离散几何结构的联系导致与材料科学问题的联系。涉及ZETA功能和素数分布的数量理论中的问题与数学物理学的问题相似。 数字理论与物理理论（例如通过模块化形式）有进一步的联系。数字理论结构出现在物理模型中一些确切的相变模型中。该提案调查了田野之间的几个可能的接触领域，这可能会导致与物理学和物质科学领域的研究人员的融合。赠款将支持在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。第一个和主要主题继续研究LERCH ZETA函数，该功能是三个变量的函数，该函数在专业变量上产生了Hurwitz Zeta函数和Riemann Zeta函数。与海森伯格集团及相关群体的代表理论建立了联系，该项目旨在将其更紧密地与各个群体上的自动形式形式联系起来。第二个主题涉及两个探索性项目；一个人涉及对称群体的某个虚拟表示的家族，该家族是多项式分裂特性的退化限制（Modulo P），与多项式判别基因座的性质相关。它询问是否存在潜在的几何结构，该结构可能与相关的动力学系统解释了观察到的极限特性。另一个探索性项目研究多结模块形式，该形式具有模块化不变性的函数，被拉普拉斯式的力量消灭。第三个主题是调查Riemann表面上的圆形包装，这是离散几何形状领域的主题。它考虑了在两个方向上的缩放限制的公式，一个复杂的变量极限和二磷酸近似限制。是否可以精确地完成此操作，提出了特殊问题来研究这些领域，包括研究具有有限数量圆圈的刚性包装的表面。