- Group theory and generalisations
- Linear and multilinear algebra, matrix theory
- Non-associative rings and algebras
Linear algebraic groups are matrix groups defined by polynomials. In the past century, a lot of research has been done to develop a classification of these algebraic groups. Among the objects of most interest in this theory are the exceptional groups. Though their classification is complete, a lot of questions remain about these mysterious objects.
Recently, a class of algebras that have these exceptional groups as symmetries have been discovered. We aim to describe these algebras in an independent manner, without referring to their symmetries, as well as construct analogues of this class of algebras.
These results ought to be fundamental to the understanding of the exceptional groups, as they would provide a way of describing them as symmetries of an a priori completely independent object. In the smallest cases we already have some examples of this, namely the octonion algebras and the Albert algebras, both extremely important algebras in their own right. This suggests that the algebras we are studying ought to be fundamental objects in the mathematical landscape.