Mathematical Sciences: Arithmetic Representation by Integral Quadratic Forms

数学科学：积分二次形式的算术表示

• 批准号: 9500533
• 负责人: Donald James
• 金额: $ 5.37万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500533&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Representation; Integral

This award supports mathematical research involving quadratic and hermitian forms defined over the integers of algebraic number fields, their connections in algebraic geometry, and Fuchsian groups and the related hyperbolic geometry. There are two specific goals of this research. The first is to determine necessary and sufficient conditions for the local primitive representation of one form by another integral quadratic form. The second is to study analogues for hermitian forms of Milnor's theorem on the classification and structure of unimodular indefinite quadratic forms over the rational integers. This research falls into the general mathematical area 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.

该奖项支持数学研究，包括代数数域整数上定义的二次和厄米特形式，它们在代数几何中的联系，以及Fuchsian群和相关的双曲几何。这项研究有两个具体目标。第一个是确定一个形式的局部本原表示为另一个积分二次型的充分必要条件。第二是研究Milnor定理关于有理整数上的单模不定二次型的分类和结构的埃尔米特形式的类似物。 这一研究属于数论的一般数学领域，属于福尔斯的研究范畴。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。它是数学最古老的分支之一，几个世纪以来出于纯粹的美学原因而受到人们的追求。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。