The Structure of Permutation Classes

排列是最基本、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是必不可少的：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。在许多这些理论中，排列的序理论性质是最重要的，这方面的抽象理论被称为排列模式理论。从Knuth的《计算机程序设计艺术》中关于堆栈排序的一个小话题开始，这个理论在过去的二十年里经历了一个快速扩张和发展的时期。PI和他的两位合作者Albert和Vatter为这一发展做出了一些开创性的贡献，他们始终强调该领域的理论/结构基础，与数学其他部分的联系，以及普遍的适用性。在过去的一年里，这种努力使团队意识到所谓的网格分解以及表示模式类的相关几何方法的至关重要性。有强有力的证据表明，这与现有的单词和形式语言编码理论相结合，将为模式类提供一个全面的结构理论，并为解决几个突出的开放问题提供一条途径。

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.