Mathematical Sciences: Functions of Binary Variables

该奖项支持罗格斯大学Peter Hammer教授在组合学和运筹学方面的研究。哈默博士的工作集中在伪布尔函数的研究上，这种函数通常将一个数字与有限集合的每个子集联系起来。在科学上的应用也很多；仅举几个应用，这项工作对最小化有影响，如VLSI设计，以及物理学中的Ising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。并辅以高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，这是当今数学中最活跃的领域之一。从根本上说，组合学代表了对所有数学活动——计数——的系统化。然而，在其现代发展中，组合学已经不仅仅是计算，而是利用了各种各样的先进数学技术，尽管它的根源可以追溯到几个世纪以前，但由于它在通信和信息技术中的重要性，该领域在过去的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.