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

图、设计、代码和群：代数组合主题

• 批准号: RGPIN-2016-05397
• 负责人: Bailey, Robert
• 金额: $ 1.31万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=755177
• 关键词: Graphs; Designs; Codes; Groups; Topics

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.

该提案的标题反映了我对被称为代数组合学的离散数学分支的广泛主题的利益，主要是在图理论和组合设计理论中，但也与有限的群体理论和编码理论有关。这些主题具有广泛的吸引力。 图理论的应用与计算机网络的建模或有机化合物（如长链聚合物）一样多样化。 设计理论起源于统计实验的设计和锦标赛计划。 对有限置换组的研究是抽象代数中最古老的主题之一，是对称概念的数学抽象，但在减少计算问题解决的难度方面非常相关。 编码理论是对数据的准确传输和存储的数学研究。 这些区域中的每个区域之间也有一个显着的重叠。该提案具有三个组成部分。 第一部分涉及图形和组，研究图形的度量尺寸（使用距离有效定位顶点，类似于智能手机确定其地理位置的方式）和置换组的相关参数。 第二部分是构建一个在线数据库的距离定期和强烈规则的图表，这应该成为数学研究界的宝贵资源。 第三部分涉及组合设计理论，开发了广义包装和覆盖设计的理论，该理论提供了一个共同的框架，可以同时了解各种不同的组合设计家族，并在编码，通信和软件测试等领域中适用于该领域。提案项目的三个组成部分都可以培训高度培训的人，以培训高级人士的培训。