An algebraic approach to the Erdos-Ko-Rado theorem

设计理论和极值集合理论是研究满足某些属性集合（也许集合必须以特定的模式相交，或者集合不能互为子集）的集合系统（和其他组合对象）。这一领域的研究通常包括寻找特定类型系统的结构，以及确定具有特定限制的最大系统的大小和结构。这是一个重要的研究领域，因为这些设计经常在应用中使用。例如，我研究的一个设计涵盖了用于为软件和网络设计有效测试套件的阵列。由于目前解决此类问题的方法通常是临时的，因此本研究计划的一个关键主题是利用代数组合学和代数图论的工具开发一种更系统的方法来解决这些类型的问题。我的研究还引入了表征理论、群论和计算机算法的元素。我目前的工作是围绕着可能是极值集理论中最著名的定理，Erdos-Ko-Rado定理。这个优雅的定理给出了由n个元素组成的基本集合中大小为k的相交子集组成的最大系统的大小，并准确地描述了哪些系统满足这个界。这个定理有许多不同的证明、推广、扩展和应用。我对这个定理的推广特别感兴趣，它适用于集合以外的对象，并且有代数证明。

Design theory and extremal set theory are the study of systems of sets (and other combinatorial objects) that satisfy some collection of properties (perhaps the sets must intersect in specific patterns or maybe the sets cannot be subsets of one another). This field of research typically includes finding constructions for specific types of systems and the determination of the size and structure of the maximum systems with a specific restriction. This is an important research area since these designs are often used in applications. For example, one design I study are covering arrays which are used to design effective test suites for software and networks. Because the current approaches to such problems are often ad hoc, a key theme of this research program is to develop a more systematic approach to these types of problems using tools from algebraic combinatorics and algebraic graph theory. My research has also brought into play elements from representation theory, group theory and computer algorithms.My current work is centered around what is probably the most famous theorem in extremal set theory, the Erdos-Ko-Rado theorem. This elegant theorem gives the size of the largest system of intersecting subsets of size k from a base set of n elements and describes exactly which systems meet this bound. There are many different proofs, generalizations, extensions and applications of this theorem. I am particularly interested in the generalizations of this theorem that apply to objects other than sets, and which have an algebraic proof.