Mathematical Sciences: Investigations in Number Theory

该奖项支持数论三个领域的研究。第一个问题涉及到小高度的数字与它们所生成的字段的算术属性之间的关系。这可能会产生一些深入了解莱默猜想，以及到现象，即权力的小塞勒姆号码往往变成特殊的单位。第二个项目是研究一个整数模m的乘法阶的变化为不同的m，并调查这些结果的推广数域，功能领域，和阿贝尔品种。第三个项目涉及动力系统的算术理论。在这方面正在研究的问题包括合理周期点的周期界限，轨道积分点的有限性，合理映射的模域，以及正则高度zeta函数的解析延拓。 这项研究是在数论的一般领域。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。

This award supports research in three areas of number theory. The first involves the relationship between numbers of small height and the arithmetic properties of the fields that they generate. This may yield some insight into Lehmer's conjecture, as well as into the phenomenon that powers of small Salem numbers frequently turn out to be exceptional units. The second project is to study the variation of the multiplicative order of an integer modulo m for varying m, and to investigate generalizations of these results to number fields, function fields, and abelian varieties. The third project concerns the arithmetic theory of dynamical systems. Problems being studied in this area include bounds for periods of rational periodic points, finiteness of integral points in orbits, fields of moduli for rational maps, and analytic continuation of canonical height zeta functions. This research is in the general 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.