Combinatorics and Number Theory II

近年来，数论在其他领域的应用越来越引起人们的兴趣。这项调查将利用数论和组合数学之间的丰富的相互作用。要调查的问题要么有他们的起源在组合数学或有直接的应用组合数学，但问题本身是数论。大致说来，本研究的问题可以分为两个部分。第一部分是最近与C. L. Chai，关于函数域上一般线性群的自守形式的显式构造。这样的形式将自动满足广义拉马努金猜想。关于某些Ramanujan图的特征值为Fourier系数的自守形式的Sato-Tate猜想有望得到证明。在这一部分中还提出了Ramanujan超图的研究，他们与建筑物，超图的特征值和Hecke算子的特征值之间的联系。其目的是了解在何种程度上拉马努金图和经典的尖点形式之间的连接可以推广到更高的秩群。第二部分是数论在良码构造中的应用。所提出的方法包括p-adic分析，特征和估计，和显式构造的无限塔的曲线与许多合理的点。PI计划写一个续集她的书“数论与应用”，包括结果之间的相互作用数论和特征值的拉马努金图在最近几年以及新的结果，从这个建议。 这本书的内容也将在她的家乡机构的研究生课程中教授。这项研究福尔斯属于数论和组合学的一般数学领域。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统，这是现代信息理论的中心主题。组合数学也是现代通信中不可或缺的工具。组合数学的目标之一是找到有效的方法来研究如何安排对象的离散集合。大型网络的设计，如电话系统中的网络，卫星和无线通信中使用的良好纠错码，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要利用数论技术进行组合研究。

The applications of Number Theory to other fields have been attracting increasing interest in recent years. This investigation will exploit the rich interplay between Number Theory and Combinatorics. The problems to be investigated either have their origin in Combinatorics or have immediate application to Combinatorics, yet the problems themselves are Number- Theoretic. Roughly speaking, the problems in this investigation can be divided into two parts. The first part is an extension of the recent joint work with C.-L. Chai, concerning explicit constructions of automorphic froms for general linear groups over function fields from geometric setting. Such forms will automatically satisfy the generalized Ramanujan conjecture. The Sato-Tate conjecture for the automorphic forms whose Fourier coefficients are eigenvalues of certain Ramanujan graphs is expected to be proved. Also proposed in this part is the study of Ramanujan hypergraphs, their connection with buildings, and the connection between the eigenvalues of hypergraphs and eigenvalues of Hecke operators. The purpose is to understand to which extent the connection between Ramanujan graphs and classical cusp forms can be generalized to higher rank groups. The second part is applications of number theory to the construction of good codes. The proposed approaches include p-adic analysis, character sum estimates, and explicit constructions of infinite towers of curves with many rational points. The PI plans to write a sequal to her book "Number Theory with Applications", including results on the interaction between number theory and eigenvalues of Ramanujan graphs obtained in recent years as well as new results from this proposal. The content of this book will also be taught in a graduate course at her home institution.This research falls into the general mathematical fields of Number Theory and Combinatorics. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems, which is the central theme of modern information theory. Combinatorics is also an indispensable tool in modern communications. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The design of large networks, such as those occurring in telephone systems, good error-correcting codes used in satellite and wireless communications, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research using techniques from number theory.