Arithmetic Ramsey theory, graph theory and combinatorics

在当今快速发展的技术世界中，许多发展都是纯数学中旧工作的结果。例如，许多现代加密技术都是基于17世纪的数论，直到最近，这些思想似乎只不过是计数的奇怪性质。作为另一个例子，现代运输或通信网络目前使用来自所谓的“图论”的基本思想来分析和优化，图论是对使用点和连接点的线形成的结构的性质的研究。我目前的研究计划考虑了许多这类应用的基础上的纯数学，但我把“应用”留给其他专家。 我正试图发现更多我认为是美丽的性质的图和数论独立于如何利用它们。数论和图论的对称性和内在的“美”，以及它们之间的关系，构成了我研究的主体。我计划研究的另一个领域类似于图论，称为格理论，它研究具有更多“秩序”和对称性的特殊类型的网络。

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.