Mathematical Sciences: Investigations in Number Theory

该奖项支持数论的三个领域的研究。第一个涉及小高度数与它们所生成的场的算术属性之间的关系。这可能会对莱默猜想以及小塞勒姆数的幂经常被证明是特殊单位的现象有一些洞察。第二个项目是研究模m的整数的乘法阶随m的变化，并研究这些结果在数域、函数域和交换变种上的推广。第三个项目涉及动力系统的算术理论。这一领域正在研究的问题包括有理周期点的周期的界，轨道上积分点的有限性，有理映射的模域，以及标准高度Zeta函数的解析延拓。这项研究属于数论的一般领域。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 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.