Graphs, Designs, Codes and Groups: Topics in Algebraic Combinatorics

这个提案的标题反映了我对离散数学分支代数组合学的广泛主题的兴趣，主要是图论和组合设计理论，但也与有限群论和编码理论有关。******这些话题具有广泛的吸引力。图论的应用非常广泛，如计算机网络的建模，或有机化合物，如长链聚合物。设计理论起源于统计实验的设计和比赛的安排。有限置换群的研究是抽象代数中最古老的学科之一，是对称概念的数学抽象，但它与降低计算问题解决的难度非常相关。编码理论是对数据准确传输和存储的数学研究。这些领域之间也有明显的重叠。******这个建议有三个组成部分。第一部分涉及图和组，研究图的度量维度（使用距离有效地定位顶点，类似于智能手机如何确定其地理位置）和排列组的相关参数。第二部分是建立距离正则图和强正则图的在线数据库，这将成为数学研究界的宝贵资源。第三部分涉及组合设计理论，发展了广义包装和覆盖设计的理论，它提供了一个共同的框架，在这个框架中，可以同时理解各种不同的组合设计家族，以及在编码、通信和软件测试等领域的应用。******提案的所有三个组成部分都具有非常适合培养高素质人才的项目，从本科生到博士后研究人员

The title of this proposal reflects my interests in a broad range of topics in the branch of discrete mathematics known as algebraic combinatorics, primarily in graph theory and combinatorial design theory, but also in connection with finite group theory and coding theory.******These topics have widespread appeal. Graph theory has applications as diverse as the modelling of computer networks, or organic chemical compounds such as long-chain polymers. Design theory originated in the design of statistical experiments and the scheduling of tournaments. The study of finite permutation groups is one of the oldest subjects in abstract algebra, being a mathematical abstraction of the notion of symmetry, but one which is very relevant in reducing the difficulty of computational problem solving. Coding theory is the mathematical study of the accurate transmission and storage of data. There is also a significant overlap between each of these areas.******This proposal has three components. The first part concerns graphs and groups, studying the metric dimension of graphs (efficiently locating vertices using distances, similar to how a smartphone determines its geographical location) and related parameters for permutation groups. The second part is to build an online database of distance-regular and strongly regular graphs, which should become a valuable resource for the mathematics research community. The third part concerns combinatorial design theory, developing the theory of generalized packing and covering designs, which provide a common framework in which to understand a variety of different families of combinatorial designs simultaneously, as well as having applications to areas such as coding, communications and software testing.******All three components of the proposal feature projects well-suited to the training of highly qualified personnel, from undergraduate students up to postdoctoral research fellows.**