Q-polynomial schemes, coherent configurations, and applications

组合数学是一个广泛的数学领域，它已经应用于许多其他领域，如计算机科学，统计学，物理学和化学。关联方案和相干配置为组合学的几个领域提供了一个统一的框架，如编码理论，实验的统计设计和有限几何。这项工作有可能阐明许多问题，在其他领域的组合，包括上述以及极值图论。这个项目将进一步探索代数和组合学之间的丰富联系，并帮助划分新的方向，问题和问题，从而激发对该领域的进一步兴趣。更广泛的影响包括锐化数学工具在工业中的应用，为学术界和工业界培养高素质的研究生，以及本科生的研究机会。线性和抽象代数与组合学之间的相互作用一直是一个非常富有成果的研究领域，并继续在物理学，计算机科学和统计学中找到超越纯数学的应用。在这个项目中，PI和他的团队研究关联方案和相干配置。该项目的第一部分是研究具有所谓的Q-多项式性质的关联方案，该性质正式对偶于距离正则图的概念。虽然距离正则图已经得到了很好的研究，但直到最近十年，很少有人注意到距离正则图中没有出现的Q-多项式性质。最近的结果表明，这些对象本身是感兴趣的，这些计划的令人惊讶的结构值得进一步探索。PI将继续寻找Q-多项式方案的新例子，特别强调那些来自群体的方案。搜索将补充工作的特点Q-多项式计划。该项目的第二部分涉及扩展关联方案的结果，更一般的概念，连贯的配置，一个自然的概括的关联方案。特别是，PI将探索最近发现的半定界在相干配置的进一步应用。

Combinatorics is a broad area of mathematics that has found applications to many other fields such as computer science, statistics, physics, and chemistry. Association schemes and coherent configurations give a unified framework for several areas of combinatorics, such as coding theory, the statistical design of experiments, and finite geometry. This work has the potential to shed light on many problems in other areas of combinatorics, including those mentioned above as well as extremal graph theory. This project will further explore the rich connections between algebra and combinatorics, and help demarcate new directions, problems, and questions, thereby stimulating further interest in the area. Broader impacts include a sharpening of mathematical tools for applications in industry, training of highly qualified graduate students for academia and industry, and undergraduate research opportunities. The interaction between linear and abstract algebra and combinatorics has been a very fruitful area of study and continues to find applications beyond pure mathematics, in physics, computer science, and statistics. In this project, the PI and his team study association schemes and coherent configurations. The first part of the project is a study of association schemes with the so-called Q-polynomial property, a property formally dual to the notion of a distance-regular graph. While distance-regular graphs have been well studied, until the last decade little attention was paid to schemes with the Q-polynomial property that did not also arise from distance-regular graphs. Recent results suggest that these objects are of interest in and of themselves, and that the surprising structure of these schemes merits further exploration. The PI will continue the search for new examples of Q-polynomial schemes, with particular emphasis on those that arise from groups. The search will be complemented by work to characterize Q-polynomial schemes. The second part of the project concerns extending results of association schemes to the more general notion of coherent configurations, a natural generalization of association schemes. In particular, the PI will explore further applications of the recently discovered semidefinite bound in coherent configurations.