Arithmetic Ramsey theory, graph theory and combinatorics

算术拉姆齐理论、图论和组合学

• 批准号: 228064-2009
• 负责人: Gunderson, David
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2010
• 资助国家: 加拿大
• 起止时间: 2010-01-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=454736
• 关键词: Arithmetic; Ramsey; theory; graph; combinatorics

In today's fast growing world of technology, many developments have been the result of old work done in pure mathematics. For example, many modern encryption techniques are based on number theory done in the 1600's, ideas that until recently seemed to be not much more than curious properties of the counting numbers. As another example, modern transportation or communication networks are presently analyzed and optimized using basic ideas from what is known as "graph theory", the study of properties of structures formed using dots and lines connecting the dots. My present research proposal considers the pure mathematics underlying many of these kind of applications, however I leave the "applying" to other specialists. I am trying to discover more of what I consider to be beautiful properties of graphs and number theory independent of how they are utilized. It is the symmetry and inherent "beauty" of both number theory and graph theory, and how they relate, that forms the bulk of my research. Another area I plan to study is similar to graph theory, called lattice theory, which studies special kinds of networks that have more "order" and symmetry.

在当今快速发展的技术世界中，许多发展都是纯数学完成的旧工作的结果。例如，许多现代化的加密技术基于1600年代所做的数字理论，直到最近似乎只是计数数字的奇怪属性。作为另一个例子，现代运输或通信网络目前是使用所谓的“图形”，即使用连接点的点和线形成的结构的属性的研究来分析和优化的。我目前的研究建议考虑了许多此类应用程序的纯数学，但是我将“申请”留给其他专家。 我试图发现更多我认为是图形的美丽属性，而数字理论则独立于使用它们的方式。数字理论和图理论的对称和固有的“美”及其如何联系，构成了我的大部分研究。我计划研究的另一个领域类似于图理论，称为晶格理论，该理论研究了具有更多“顺序”和对称性的特殊网络。