Rigorous Experimental Combinatorics

严格的实验组合学

• 批准号: 0901226
• 负责人: Doron Zeilberger
• 金额: $ 28.75万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901226&HistoricalAwards=false
• 关键词: Rigorous; Experimental; Combinatorics

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.

Doron Zeilberger建议继续开发利用计算机代数的巨大潜力进行组合学和相关领域研究的方法，并为（严格的）计算机辅助和计算机生成的研究设计实验。他特别希望在排列统计理论中开发一个新的、计算机辅助和计算机生成的阶段;从符号计算而不是数字计算的角度探索组合统计物理中看似棘手的问题;研究格路径枚举，并扩展Wilf-Zeilberger算法证明理论以处理无限多个变量。他还希望解决关于所谓的完全填充循环的有趣的Razumov-Stroganov猜想，即与交替符号矩阵的简单双射。这项研究应该是共生的，因为预计具体结果和底层方法都将帮助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，因为它将有助于开发一种研究方法，用于进行计算机实验，输出严格（和有趣！）数学定理（和证明），而不仅仅是验证和制定公式。这项研究是在组合数学领域，其有用的科学和技术是众所周知的。特别是，计算机科学在很大程度上是基于组合学，电子通信和万维网也是如此。