Rigorous Experimental Combinatorics

Doron Zeilberger建议继续开发方法，利用计算机代数的巨大潜力，在组合学和相关领域进行研究，并为（严格的）计算机辅助和计算机生成研究设计实验。他特别希望在置换统计理论中发展一个新的、计算机辅助和计算机生成的阶段；从符号计算而非数值计算的角度探讨组合统计物理中看似棘手的问题；研究点阵路径枚举，并将Wilf-Zeilberger算法证明理论扩展到处理无限多变量。他还希望解决关于所谓的完全填充环的有趣的Razumov-Stroganov猜想，即具有交替符号矩阵的简单双射。这项研究应该是共生的，因为预计具体的结果和潜在的方法，将有助于计算机代数开发人员改进和增强他们的系统。也希望这项研究将有助于实验数学的萌芽领域，因为它将有助于发展一种进行计算机实验的研究方法，这些实验可以输出严格（和有趣的！）数学定理（和证明），而不仅仅是验证和制定猜想。这项研究属于组合学领域，其对科学技术的用处是众所周知的。特别是，计算机科学在很大程度上建立在组合学的基础上，电子通信和万维网也是如此。

Doron Zeilberger proposes to continue to develop methodologies for harnessing the great potential of Computer Algebra to do research in Combinatorics and related areas, and design experiments for (rigorous) computer-assisted and computer-generated research. In particular he hopes to develop a new, computer-assisted and computer-generated phase, in the theory of permutation statistics; explore seemingly intractable problems in combinatorial statistical physics from the point of view of symbolic, rather than numeric, computations; study lattice paths enumeration, and extend the Wilf-Zeilberger algorithmic proof theory to handle indefinitely many variables. He also hopes to tackle the intriguing Razumov-Stroganov conjecture about so-called Fully Packed Loops, that are in simple bijection with Alternating Sign Matrices.This research should be symbiotic, as it is expected that both the concrete results and the underlying methodologies, would help computer algebra developers to improve and enhance their systems. It is also hoped that this research will contribute to the budding field of Experimental Mathematics, in that it will help develop a research methodology for conducting computer experiments that output rigorous (and interesting!) mathematical theorems (and proofs), rather than just verifying and formulating conjectures.This research is in the field of Combinatorics, whose usefulness to science and technology is well-known. In particular, computer science is largely based on combinatorics, as is electronic communication and the World Wide Web.