International Workshop Nonassociative Algebras and Geometry

The workshop will focus on the theory of nonassociative algebras and their applications. More precisely, we plan to discuss the following areas:

Exceptional Lie groups and algebraic groups.

Nonassociative algebras play an important role in the classification theory of simple Lie algebras, algebraic groups, and homogeneous spaces. For example, one can mention the celebrated Freudenthal magic square and the Tits construction.

Gradings by arbitrary groups.

Gradings on nonassociative algebras have proved useful in the theory of simple Lie algebras. For instance, gradings on Hurwitz algebras play a decisive role in the classification of gradings on exceptional Lie algebras.

Polynomial identities.

It is a well-known stumbling block in the theory of PI-algebras that identities of matrix algebras are unknown beyond matrices of size 2. Graded polynomial identities are much more tangible. On the other hand, two (nonassociative) algebras with the same graded identities also have the same ordinary identities.

Simple nonassociative algebras.

The classification of finite-dimensional nonassociative central simple or division algebras over an arbitrary field is an important problem, similar to the determination of the Brauer group of the field. This requires techniques from various branches of mathematics, in particular, from geometry and topology.

Geometry and algebraic cycles of exceptional flag varieties.

The theory of versal torsors and algebraic cycles on twisted flag varieties has been recently developed by Karpenko and has connections to the theory of linear algebraic groups over arbitrary fields and quadratic forms. The study of versal torsors for exceptional flag varieties of types G2 and F4 is a new direction of research, which could lead to a better understanding of their geometries.

Nonassociative algebras in physics.

The approach to the standard model of elementary particles based on noncommutative geometry, proposed by Alain Connes, dates back to the late nineties and early two-thousands. More recently, nonassociative algebras have been used as a model for the foundations of quantum mechanics, as a model for magnetic monopoles, and for quantum gravity. One goal of the workshop is to connect the mathematics community and the physics community that both work on these questions from different sides.