Zeta Integrals, Discrete Number Theory and Geometry

这个研究项目涉及数论和几何学之间的相互作用，并与物理学有联系。数论是离散的，几何是连续的，但连续结构可以通过越来越精细的近似被离散结构所近似。中间离散模型可以同时显示（离散）数论和（连续）几何性质。通过这种方式，数论通过封装与离散几何结构联系起来，从而与材料科学中的问题联系起来。某些大尺寸的有限离散模型与物理学中精确可解的模型有相似之处，当温度参数发生变化时，它们表现出相变，比如在水和冰之间。本提案中的项目研究了与这些类比相关的一些具体问题，以便在各种限制下架起离散和连续的桥梁。这项工作将导致与几何、物理和材料科学研究人员富有成效的互动，并将支持研究生在数论和离散几何方面的培训。更详细地说，PI将调查与数论和几何有关的几个主题。第一个，也是主要的，主题继续研究莱赫zeta函数，这是一个三变量的函数，在专门化变量上产生Hurwitz zeta函数和Riemann zeta函数。这项研究是基于最近发现的这些函数与真实海森堡群和相关可解群的表示之间的联系。该项目将研究通过zeta积分将lach zeta函数推广到各种高维李群的自同构表示。第二个主题是研究算术结构的各种离散有限模型，通常有两个变量，以及随着模型大小的增加对这些模型的统计研究，寻求与统计力学中的可积系统的类比。第三个主题是望月提出的宇宙内Teichmuller理论。这个新理论将从一个更容易理解的分析角度重新表述。第四个主题是关于黎曼曲面上的圆填充。PI将在两个方向上研究完美有限圆填料的标度极限：复变量极限和丢番图近似极限。这些限制将被精确地确定。它们之间的关系也将被确定。

This research project concerns interactions between number theory and geometry, with connections to physics. Number theory is discrete and geometry is continuous, but continuous structures can be fruitfully approximated by discrete structures through finer and finer approximations. Intermediate discrete models can exhibit both (discrete) number theoretic and (continuous) geometric properties. In this way, number theory connects with discrete geometric structures, via packings, leading to connections with problems in materials science. Certain finite discrete models in large sizes have parallels with exactly solvable models in physics that exhibit phase transitions when a temperature parameter is varied, as between water and ice. The projects in this proposal investigate a number of specific problems related to these analogies in order to bridge the discrete and the continuous in various limits. This work will lead to fruitful interactions with researchers in geometry, physics and material science and will support the training of graduate students in number theory and in discrete geometry.In more detail, the PI will investigate several topics relating number theory and geometry. 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 both the Hurwitz zeta function and the Riemann zeta function. This research is based on a recently discovered connection between these functions and representations of the real Heisenberg group and related solvable groups. The project will study generalizations of the Lerch zeta function to automorphic representations of various higher-dimensional Lie groups, via zeta integrals. A second topic concerns the study of various discrete finite models of arithmetic structures, typically with two variables, and the study of statistics of such models as their size increases, pursuing an analogy with integrable systems in statistical mechanics. A third topic concerns intra-universal Teichmuller theory introduced by Mochizuki. This new theory will be reformulated from a more accessible analytic perspective. A fourth topic concerns circle packings on Riemann surfaces. The PI will study scaling limits of perfect finite circle packings in two directions: a complex variables limit and a Diophantine approximation limit. These limits will be determined precisely. How they are related will also be determined.