Mathematical Sciences: Functions of Binary Variables

该奖项支持罗格斯大学彼得·哈默教授在组合学和运筹学方面的研究。哈默博士的工作主要集中在伪布尔函数的研究上，伪布尔函数通常将一个数字与有限集的每个子集相关联。科学上的应用很多；举几个例子，这项工作对最小化有意义，比如在VLSI设计中，以及在物理学中的伊辛自旋玻璃模型和神经网络。这个项目的具体目标是一方面更好地理解伪布尔函数，重点是建立一个连贯的理论，另一方面发展这些函数的计算数学，并伴随着有效代码的实现和数值实验。这项研究属于组合学的广泛范畴，它是当今数学中最活跃的领域之一。从根本上说，组合学代表着对所有数学活动中的第一个--计数--的系统化。然而，在它的现代发展中，组合数学已经不仅仅是利用各种先进的数学技术，尽管它的根源可以追溯到几个世纪前，但由于它在通信和信息技术中的重要性，该领域在过去几十年里有了爆炸性的发展。

This award supports the research in Combinatorics and Operations Research of Professor Peter Hammer of Rutgers University. Dr. Hammer's work centers on the study of pseudo- Boolean functions, which typically associate a number to each subset of a finite set. Applications in science are numerous; to name a few applications, this work has implications for minimization as in VLSI design, and Ising spin glass models and neural networks in physics. The specific goal of this project is on the one hand to gain a better understanding of pseudo- Boolean functions, with an emphasis on the construction of a coherent theory, and on the other hand to develop the compu- tational mathematics of these functions, accompanied by the implementation of efficient codes and by numerical experi- mentation. This research falls in the broad category of Combinatorics, which is one of the most active fields in today's mathematics. Fundamentally, Combinatorics represents a systematization of the very first of all mathematical activities, counting. In its modern development, however, Combinatorics has gone beyond just counting to make use of a wide variety of advanced mathematical techniques, and although its roots go back several centuries, the field has had an explosive development in the past few decades because of its importance in communications and information technology.