Combinatorics and Number Theory II

近年来，数论在其他领域的应用引起了人们越来越大的兴趣。这项调查将利用数论和组合学之间丰富的相互作用。所要研究的问题要么起源于组合学，要么直接应用于组合学，但问题本身是数论的。粗略地说，这次调查中的问题可以分为两个部分。第一部分是最近与C-L.Chai合作工作的推广，涉及从几何设置到函数域上一般线性群的自同构形的显式构造。这样的形式将自动满足广义Ramanujan猜想。证明了某些Ramanujan图的傅里叶系数为特征值的自同构型的Sato-Tate猜想。这部分还提出了Ramanujan超图的研究，它们与建筑物的联系，以及超图的特征值与Hecke算子的特征值之间的联系。其目的是了解Ramanujan图和经典尖点形式之间的联系在多大程度上可以推广到更高阶群。第二部分是数论在构造良码中的应用。所提出的方法包括p-进分析、特征和估计以及具有多个有理点的无限曲线塔的显式构造。PI计划撰写她的《数论与应用》一书的续集，包括近年来在数论与Ramanujan图的特征值之间的相互作用方面的结果，以及这一提议的新结果。这本书的内容也将在她的家乡学院的研究生课程中教授。这项研究属于数论和组合学的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为现代信息理论的中心主题--数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。组合学也是现代通信中不可或缺的工具。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。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的纠错码，以及计算机科学中处理离散对象集的算法的设计，这些都利用了使用数论技术的组合研究。

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.