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.

Combinatorics是数学的广泛领域，它已在许多其他领域（例如计算机科学，统计，物理和化学）找到了应用。关联方案和连贯的配置为组合学的多个领域提供了一个统一的框架，例如编码理论，实验的统计设计和有限的几何形状。这项工作有可能阐明组合学领域的许多问题，包括上述问题以及极端图理论。该项目将进一步探索代数与组合学之间的丰富联系，并帮助划定新的方向，问题和问题，从而刺激对该地区的进一步兴趣。更广泛的影响包括为行业应用的数学工具增强，对学术界和工业的高素质研究生的培训以及本科研究机会。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，并且在物理，计算机科学和统计学中继续发现超出纯数学的应用。在这个项目中，PI和他的团队学习协会计划和连贯的配置。该项目的第一部分是对与所谓的Q-Polynomial特性的关联方案的研究，该属性是正式双重距离距离规则图的属性。虽然对距离规则的图进行了充分的研究，但直到最近十年对具有Q-Polynomial特性的方案的关注很少，这也不是距离定量图。最近的结果表明，这些对象本身就是感兴趣的，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-Polynomial方案的新示例，并特别强调了由小组产生的示例。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将探索以相干配置结合的最近发现的半芬矿的进一步应用。