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.
The main aim was to present the applications of nonassociative algebras to other areas that have occurred in the past thirty years to a new generation of researchers. Some of these applictions were quite profound. For example, Griess had used nonassociative algebras in his construction of the Monster finite simple group, one of the most famous finite structures in algebra ever discovered. Zelmanov discussed the role of nonassociative algebras in his solution to the famous Burnside problem, at the interface of the realms of the finite and infinite, that had won him the Fields medal. Shestakov and a collaborator had used the subject to crack a long open problem on polynomials in three variables. Parimala discussed the surprising relevance of nonassociative algebras in her work related to number theory. Kac discussed nonassociative algebras in the context of work on physics. Many participants mentioned in their talks the work in nonassociative algebras of Kevin McCrimmon, who was honored at a conference dinner. Many participants were young researchers, pre or post PhD. They came from areas nearby as well as national and world-wide, and contained members of many underrepresented groups in mathematics. There was ample opportunity for discussion and personal contact, and many participants expressed their appreciation of the conference format.