Combinatorics, Special Functions, and Computer Algebra

该奖项支持多伦·泽尔伯格教授在代数组合学、特殊函数理论和计算机代数领域的研究。这个项目的一个方向是继续他的研究计划，使用计算机代数来计算机化地证明组合学中的恒等式和特殊函数理论。这个项目的其他目标包括发展置换统计的一般理论，研究组合统计力学的完整方面，以及将首席研究员最近证明交替符号矩阵猜想所发展的方法扩展到这一领域的其他仍未解决的问题。由于特殊函数和组合学在自然科学中非常普遍，这项研究在科学和技术以及纯数学中都有许多潜在的应用。这项研究属于组合学的一般领域。组合学试图找到有效的方法来研究离散的对象集合如何排列。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。

This award supports the research of Professor Doron Zeilberger in the areas of algebraic combinatorics, the theory of special functions, and computer algebra. One direction of this project is a continuation of his research program on the use of computer algebra for computerized proofs of identities in combinatorics and the theory of special functions. Other goals of this project include the development of a general theory of permutation statistics, the study of holonomic aspects of combinatorial statistical mechanics, and the extension of methods developed in the principal investigator's recent proof of the alternating sign matrix conjecture to other, still open, problems in this area. Because both special functions and combinatorics are so prevalent in the natural sciences, this research has many potential applications to both science and technology, as well as to pure mathematics. This research is in the general area of combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects 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.