Mathematical Sciences: Number Theory and Allied Topics

数学科学：数论及相关主题

• 批准号: 9501101
• 负责人: George Andrews
• 金额: $ 16.03万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501101&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Number; Theory; Allied

9501101: Andrews Abstract. This award supports the research of Professors George Andrews and W. Dale Brownawell in combinatorics, number theory and commutative algebra. One main theme of this research is to develop new methods in q-series and hypergeometric series to further discoveries in the theory of partitions, number theory, representation theory and the applications of random graphs in computer science. A second direction is to study an algorithmic effective Nullstellensatz, isogeny of Drinfeld modules, independence criteria in diophantine rings, and zero estimates for solutions of differential equations. This research is in the general areas of combinatorics and number theory, and also involves the interaction of these areas with commutative algebra and algebraic geometry. Combinatorics attempts to find efficient methods to study how discrete collections of objets can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. 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.

9501101：安德鲁斯摘要。该奖项支持乔治·安德鲁斯教授和W·戴尔·布朗威尔教授在组合学、数论和交换代数方面的研究。这项研究的一个主要主题是发展Q-级数和超几何级数的新方法，以进一步发现分割论、数论、表示论以及随机图在计算机科学中的应用。第二个方向是研究算法有效的Nullstellensatz，Drinfeld模的同源性，丢番图环上的独立性准则，以及微分方程解的零估计。这项研究是在组合学和数论的一般领域，也涉及这些领域与交换代数和代数几何的相互作用。组合学试图找到有效的方法来研究如何安排离散的对象集合。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。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。