Q-polynomial schemes, coherent configurations, and applications

Q 多项式方案、相干配置和应用

• 批准号: 1400281
• 负责人: Jason Williford
• 金额: $ 22.81万
• 依托单位: University of Wyoming
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400281&HistoricalAwards=false
• 关键词: polynomial; schemes; coherent; configurations; applications

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.

组合数学是数学的一个广泛领域，已应用于许多其他领域，例如计算机科学、统计学、物理和化学。关联方案和相干配置为组合数学的多个领域提供了统一的框架，例如编码理论、实验统计设计和有限几何。这项工作有可能揭示组合学其他领域的许多问题，包括上面提到的问题以及极值图论。该项目将进一步探索代数和组合学之间的丰富联系，并帮助界定新的方向、问题和疑问，从而激发对该领域的进一步兴趣。更广泛的影响包括改进工业应用数学工具、为学术界和工业界培养高素质研究生以及本科生研究机会。线性和抽象代数与组合数学之间的相互作用一直是一个非常富有成果的研究领域，并且继续在纯数学之外的物理、计算机科学和统计学中找到应用。在这个项目中，PI 和他的团队研究了关联方案和相干配置。该项目的第一部分是研究具有所谓 Q 多项式属性的关联方案，该属性在形式上与距离正则图的概念对偶。虽然距离正则图已经得到了很好的研究，但直到最近十年，人们很少关注具有 Q 多项式性质的方案，而这种性质也不是由距离正则图产生的。最近的结果表明，这些物体本身就很有趣，并且这些方案的令人惊讶的结构值得进一步探索。 PI 将继续寻找 Q 多项式方案的新示例，特别是来自群的示例。该搜索将通过表征 Q 多项式方案的工作来补充。该项目的第二部分涉及将关联方案的结果扩展到更一般的相干配置概念，即关联方案的自然概括。特别是，PI 将探索最近发现的半定界在相干配置中的进一步应用。