Mathematical Sciences: Arithmetic Representation by Integral Quadratic Forms

该奖项支持涉及定义在代数数域的整数上的二次型和厄米特型的数学研究，它们在代数几何中的联系，以及富克斯群和相关的双曲几何。这项研究有两个具体目标。首先确定了一种形式用另一种积分二次型表示的局部本原表示的充要条件。第二个是研究有理整数上单模不定二次型的分类和结构的Milnor定理的Hermite形式的类似。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。

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.