Mathematical Sciences: Stark's Conjectures on Number Fields and Theory Fields

海斯·斯塔克9626381的猜想为类场理论提供了一个重要的新视角。它们是对著名的希尔伯特第12题提出的问题的巧妙和极具原创性的回答。Hayes利用Drinfield模理论给出了函数域上的Abel-Stark猜想的构造性证明。他希望将从函数字段设置中获得的见解应用到数字字段中。Hayes已经证明，满足Leopoldt猜想的数域上的Brumer-Stark单位可以对齐成一个Hecke特征标。在这个项目中，他将试图证明更多类似于A.Weil介绍的著名的Jacobi Sum特征的Hecke特征的存在。这种性质的存在与Brumer-Stark公理在基数变化下的不变性密切相关，这一点尚未得到证明。由于斯塔克的猜想是基于零点的部分Zeta函数的特定值，理解这种特定值的含义是很重要的。Haye给出了实二次数域上这一特定值的一种解释。他将尝试将这种解释扩展到负整数的特定值和任意全实数域。最后，Hayes还将利用从与超椭圆函数场相关的Drinfield模的计算中获得的见解，继续他对一阶Drinfield模的工作。这项研究属于数论的一般数学领域。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。

Hayes 9626381 Stark's conjectures provide a major new insight into class field theory. They are an ingenious and highly original answer to the questions raised by the famous 12th problem of Hilbert. Hayes has given constructive proofs of the abelian Stark conjectures in function fields using the theory of Drinfield modules. He hopes to apply insights gained from the function field setting to number fields. Hayes has shown that the Brumer-Stark units over a number field satisfying Leopoldt's conjecture can be aligned into a Hecke character. In this project, he will attempt to prove the existence of further Hecke characters analogous to the well-known Jacobi sum characters introduced by A. Weil. The existence of such character is intimately related to the invariance under base change of the Brumer-Stark axioms, which is as yet unproved. Since Stark's conjectures are based upon the special value of partial zeta functions at zero, it is important to understand what such special values mean. Haye has given one interpertation of this special value over a real quadratic number field. He will try to extend this interpertation to the special values at negative integers and to arbitrary totally real number fields. Finallly, Hayes will also continue his work with Drinfield modules of rank one using the insights gained from the computation of Drinfield modules associated to hyperelliptic functions fields. This research falls into the general mathematical field of Number Theory. 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.