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 is a 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.
