Arithematic Manifolds: Geodesics, Spectra and L-Functions

沃尔伯特教授将研究有关负弯曲空间的问题，特别是最短路径和高能振荡。其目标是对解析数论中的问题进行新的应用。双曲流形上的测地线具有特殊的性质。Wolpert教授将继续他与A.BasMajian在双曲曲面和三维流形上描述测地线的工作。模群的谱可以作为研究量子混沌中自同构形式甚至问题的模型。谱是现代解析数论的一个主要研究课题。包括“量子极限的唯一性”、“非算术群的谱分解”和“Ramanujan-Peterson猜想”在内的几个公开问题实际上是关于某些展开系数的大小的问题。教授将利用之前一项拨款的发现来建立对这些系数的新的、更好的估计。这是在数论的一般领域中的研究。数论的历史根源在于研究整数，考虑一个整数被另一个整数整除的问题。数论是数学中最古老的领域之一，有时也指引着整个领域的发展方向。在当今时代，它已经成为计算机科学、通信和电子安全中不可或缺的工具。

Professor Wolpert will investigate questions about negatively curved space, especially shortest paths and high-energy oscillations. The goal is to make new applications to problems in analytic number theory. Geodesics on hyperbolic manifolds have special properties. Professor Wolpert will continue his work with A. Basmajian on describing geodesics in hyperbolic surfaces and three-manifolds. The spectrum of the modular group serves as a model case for the study of automorphic forms and even questions in Quantum Chaos. The spectrum is a major subject of study in modern analytic number theory. Several open questions including "uniqueness of quantum limits," "the spectral decomposition of nonarithmetic groups," and the Ramanujan-Peterson conjecture' are actually questions about the magnitude of certain expansion coefficients. Professor will use discoveries made under a previous grant to establish new and better estimates of these coefficients.This is research in the general area of number theory. The historical roots of number theory are in the study of the whole numbers, considering questions of divisibility of one whole number by another. Number theory is one of the oldest fields in mathematics and has at times guided the direction of the development of the entire field. In current times it has served as an indispensable tool in computer science, communications, and electronic security.