Mathematical Sciences: Number Theory and Allied Topics

9501101：安德鲁斯摘要。 该奖项支持教授乔治安德鲁斯和W。Dale Brownawell在组合学，数论和交换代数。 本研究的一个主题是发展q-级数和超几何级数的新方法，以进一步发现划分理论，数论，表示论和随机图在计算机科学中的应用。 第二个方向是研究一个算法有效的Nullstellenarchy，Isarchy的Drinfeld模块，独立的标准在丢番图环，零估计的解决方案的微分方程。 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。 离散系统的行为对现代通信极为重要。 例如，大型网络的设计，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。 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。

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.