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The Structure of Permutation Classes

排列类的结构

基本信息

批准号：
EP/J006440/1
负责人：
Nik Ruskuc
金额：
$ 8.5万
依托单位：
University of St Andrews
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ006440%2F1
关键词：
Structure Permutation Classes

项目摘要

Permutations are among most fundamental and ubiquitous abstract mathematical concepts, and are indispensable in modeling a vast array of higher level phenomena in other sciences: from the study of symmetries of the material world in Physics, via genome rearrangements and evolution in Biology to the study of data processing in Computer Science. In many of these, the order-theoretic properties of permutations are of chief importance, and the abstract theory underlying this facet is known as the theory of permutation patterns. From its beginnings as a small topic in Knuth's Art of Computer Programming to do with stack sorting, the theory has undergone a period of rapid expansion and development over the past two decades. The PI and his two proposed collaborators Albert and Vatter have made several ground breaking contributions to this development, consistently placing emphasis on the theoretical/structural foundations of the field, on the links with other parts of mathematics, and on general applicability. This effort has, in the past year, lead the team to the realisation of the crucial importance of so called grid decompositions, and associated geometric ways of representing pattern classes. There is strong evidence that this, combined with the existing theory of encodings by words and formal languages, will provide both a comprehensive structure theory for pattern classes and a pathway to solving several outstanding open problems.

排列是最基本和最普遍的抽象数学概念之一，在其他科学中建模大量更高层次的现象时是不可或缺的：从物理学中物质世界的对称性研究，到生物学中的基因组重排和进化，再到计算机科学中的数据处理研究。在其中的许多理论中，排列的序论性质是最重要的，这方面的抽象理论被称为排列模式理论。从最初作为Knuth的《计算机编程艺术》中的一个小主题到与堆栈排序有关，该理论在过去的二十年中经历了一段快速扩展和发展的时期。PI和他的两个合作者Albert和Vatter为这一发展做出了一些开创性的贡献，一贯强调该领域的理论/结构基础，与数学其他部分的联系以及普遍适用性。在过去的一年里，这一努力使团队意识到所谓的网格分解的至关重要性，以及表示模式类的相关几何方法。有强有力的证据表明，这一点，结合现有的理论编码的单词和形式语言，将提供一个全面的结构理论模式类和一个途径，以解决一些悬而未决的问题。