Conference on Nonassociative Algebra in Action: Past, Present, and Future Perspectives

行动中的非结合代数会议：过去、现在和未来的观点

• 批准号: 1106203
• 负责人: Weiqiang Wang
• 金额: $ 1.3万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-01 至 2012-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106203&HistoricalAwards=false
• 关键词: Conference; Nonassociative; Algebra; Action; Past

This conference brings together researchers who have contributed to diverse and unanticipated applications of nonassociative algebras and nonassociative methods in mathematics, including Zelmanov's solution of the restricted Burnside problem using quadratic Jordan algebras , the use of Poisson brackets by Shestakov and Umirbaev in settling Nagata's conjecture on polynomial algebra symmetries, and Griess's construction of the Monster finite simple group through automorphisms of a new nonassociative algebra. Other areas of impact represented at the conference include include extended affine Lie algebras, and the arithmetic study of algebraic groups, and somewhat more classical applications include finite and continuous geometry, such as projective planes and bounded symmetric domains.To most mathematicians or physicists, Lie algebras are the most familiar examples of nonassociative algebra, known for over a hundred years as an efficient way of algebraically dealing with rules of constraint imposed by interactions in physics. Later, another type of nonassociative algebras, called Jordan algebras, was proposed as part of a possible alternative foundation for quantum physics . Soon, it was realized there were many different types of nonassociative algebras that might be worth studying, and in the 60s and 70s there was a serious study of these algebras for their own sake, an effort including mathematicians from Yale, Chicago, MIT, as well as the former Soviet Union. Though this original activity impacted mainly Lie algebras and bounded symmetric domains, the subject slowly began to have surprising new applications. This new activity took place over a period of about thirty years, beginning in the 80s, in areas not obviously related to the problems which had given it birth. While there have been, of course, conferences in the many separate areas where nonassociative algebras have been used, there have been none attempting to draw together so many of these disparate applications, and compare them for the benefit of new generations of mathematicians. It is the purpose of this conference to help these new generations and all present better understand the ways in which nonassociative algebras might be used even more broadly, to foster possible new collaborations, and to help discover the most productive directions for future research.

这次会议聚集了对非结合代数和非结合方法在数学中的各种和意想不到的应用做出贡献的研究人员，包括Zelmanov使用二次Jordan代数解决受限Burnside问题的方法，Shestakov和Umirbaev在解决Nagata关于多项式代数对称的猜想时使用的泊松括号，以及Griess通过一个新的非结合代数的自同构构造Monster有限单群。会议的其他影响领域包括扩展仿射李代数和代数群的算术研究，更经典的应用包括有限和连续几何，如射影平面和有界对称域。对于大多数数学家或物理学家来说，李代数是非结合代数最熟悉的例子，一百多年来一直被认为是用代数处理物理中相互作用施加的约束规则的有效方法。后来，另一种类型的非结合代数被提出，称为乔丹代数，作为量子物理可能的另一种基础的一部分。很快，人们意识到有许多不同类型的非结合代数可能值得研究，在60年代和70年代，人们为了这些代数本身而对它们进行了认真的研究，包括来自耶鲁、芝加哥、麻省理工学院和前苏联的数学家。虽然这个最初的活动主要影响李代数和有界对称域，但这个课题慢慢地开始有了令人惊讶的新应用。这一新的活动发生在大约30年的时间里，从80年代开始，发生在与催生它的问题没有明显关系的地区。当然，虽然在使用非结合代数的许多不同领域都有过会议，但从来没有人试图将这么多不同的应用程序聚集在一起，并为了新一代数学家的利益进行比较。这次会议的目的是帮助这些新一代和所有在座的人更好地理解非结合代数可能被更广泛地使用的方式，以促进可能的新合作，并帮助发现未来研究的最有成效的方向。