New Approaches to Maass Wave Forms in Mathematics and Physics

模曲面上拉普拉斯函数的离散谱和相关的本征函数(称为Maass波)是一类有趣的基本函数，然而在它们被发现50年后，它们仍然神秘地难以捉摸。虽然存在数值近似，但具体的分析信息有点稀缺；尤其是，目前还不存在任何这些特征函数的显式构造。这些数学对象对看似不同的领域很感兴趣，如数学和动力系统中的数论、统计力学和理论物理中的量子混沌。这个项目的起点是一类新的等同于Maass形式的函数，称为周期函数，类似于与整权的经典模形式相关的周期多项式。这些函数是由研究人员发现的，是最近与D.Zagier合作单独开发的，它们是全纯的，满足一个简单的函数方程，因此似乎是比Maass形式本身更简单的一类数学对象。因此，由于它们的代数结构的发展导致了一些不同的相互关联和有趣的方向，它们显示出继续揭示这一基本问题的新曙光的前景。利用群的表示理论和上同调，进一步发展和推广这些结果到一般的子群，是本课题的主要目标。此外，在研究动力系统的转移算符和研究统计力学中的自旋链配分函数时，周期函数和某些方程之间的联系已经被几位研究人员研究，其中包括德国的一个工作组，该工作组正在专门研究这些问题。因此，Maass周期函数在理论物理的这些相关领域的应用也是这项工作的重点，包括与致力于这些问题的研究人员的讨论。数学和相关理论物理的研究将使用研究人员和他的同事开发的方法，以获得关于Maass波形的更具体和明确的信息。这些函数是被称为“模形式”的类中的一种，在数论、物理学和密码学(或代码生成和破译)等不同领域中有许多应用。尤其是马斯波，是普通空间中的正弦波(或纯音--敲击音叉产生的波形)的非欧几里得版本，它是构成所有形状的基本成分。在这个非欧几里得的“平行宇宙”中，对几何和物理的研究为科学提供了丰富的新思想来源，以及一个对比鲜明的试验场，在这个试验场上，我们可以敏锐地研究我们生活的空间的几何和物理思想。例如，对Maass形式的行为的研究提供了非常有用的见解，以寻求更深入地理解物理系统将表现出规则或混沌行为的条件，这是物理学中一个重要而普遍的问题。事实证明，Maass形式很难明确构建，而且在发现50年后仍在进行积极的研究。

The discrete spectrum of the Laplacian on the modular surface and the associated eigenfunctions (called Maass wave forms) are an interesting and basic class of functions which nevertheless remain mysteriously elusive fifty years after their discovery. While numerical approximations exist, concrete analytic information is somewhat scarce; in particular, at present no explicit constructions exist for any of these eigenfunctions. These mathematical objects are of interest in such seemingly diverse fields as number theory in mathematics and dynamical systems, statistical mechanics, and quantum chaos in theoretical physics. The starting point for this project isa new class of functions equivalent to the Maass forms, called period functions in analogy with the period polynomials associated to the classical modular forms of integral weight. These functions, discovered by the investigator and developed individually and more recently in collaboration with D. Zagier, are holomorphic and satisfy a simple functional equation, and so appear to be simpler class of mathematical objects than the Maass forms themselves. For this reason, and because the development of their algebraic structure leads in a number of different interrelated and intriguing directions, they show promise to continue to shed new light on this basic problem. Their further development and the extension of these results to general subgroups, using representation theory and cohomology of groups, are the main goals of this project. In addition, connections between period functions and certain equations which arise both in the study of transfer operators for dynamical systems and in the study of spin-chain partition functions in statistical mechanics are already under study by several researchers, including one working group in Germany which is specifically investigating these questions. The applications of Maass period functions to these related areas of theoretical physics is thus also a focus of this work, to be pursued including discussions with researchers working on these problems. Research in mathematics and related theoretical physics will use methods developed by the investigator and his colleagues to gain more concrete and explicit information about Maass wave forms. These functions are one type in a class called "modular forms" which has many applications in such diverse fields as number theory, physics, and cryptography (or code making and breaking). Maass wave forms in particular are the non-Euclidean version of the sine waves (or pure tones - the wave form produced by striking a tuning fork) in ordinary space, which are the basic components out of which all shapes are built. The study of geometry and physics in this non-Euclidean "parallel universe" has provided science with a rich source of new ideas, as well as a contrasting proving ground on which to sharpen ideas in the geometry and physics of the space we live in. For example, the study of the behavior of Maass forms provides very useful insights in the quest to gain a deeper understanding of the conditions under which a physical system will exhibit regular or chaotic behavior, an important and pervasive question in physics. Maass forms have proven mysteriously hard to construct explicitly, and continue to be under active investigation fifty years after their discovery.