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

这次会议汇集了研究人员谁作出了贡献的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，包括Zelmanov的解决方案的限制伯恩赛德问题使用二次约旦代数，使用泊松括号的Shestakov和Umirbaev在解决永田的猜想多项式代数对称性，Griess通过一个新的非结合代数的自同构构造Monster有限单群。在会议上代表的其他领域的影响包括包括扩展仿射李代数，代数群的算术研究，和一些更经典的应用包括有限和连续几何，如投影平面和有界对称域。对于大多数数学家或物理学家，李代数是最熟悉的例子nonassociative代数，作为一种有效的代数方法来处理物理学中相互作用所施加的约束规则，它已经有一百多年的历史了。后来，另一种类型的非结合代数，称为约当代数，被提出作为量子物理学可能的替代基础的一部分。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，为了这些代数本身，人们对这些代数进行了认真的研究，这一努力包括耶鲁大学，芝加哥，麻省理工学院以及前苏联的数学家。虽然这个原始活动主要影响李代数和有界对称域，但这个主题慢慢开始有令人惊讶的新应用。这种新的活动从80年代开始，在与产生它的问题没有明显关系的领域里进行了大约30年。当然，在非结合代数被应用的许多不同领域里，也有过一些会议，但没有人试图把这么多不同的应用汇集在一起，对它们进行比较，以利于新一代的数学家。这是本次会议的目的，以帮助这些新一代和所有目前更好地理解的方式，其中非结合代数可能会被更广泛地使用，以促进可能的新的合作，并帮助发现最富有成效的方向，为未来的研究。

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.