
Natural sciences
 Group theory and generalisations ^{ }
 Linear and multilinear algebra, matrix theory ^{ }
 Nonassociative rings and algebras ^{ }
 Topological groups, Lie groups ^{ }
Nonassociative algebras play an important role in many areas of mathematics. The most prominent examples are Lie algebras, introduced in the 1930s to study infinitesimal transformations. Other classes of nonassociative algebras also proved vary fruitful in other areas; Jordan algebas, for instance, played a crucial role in Zel'manov's solution to the restricted Burnside problem in group theory. The goal of the proposed project is to explore new connections between certain classes of nonassociative algebras and group theory (and related areas). We will focus on certain vertex operator algebras (VOAs) and axial algebras. Perhaps one of the most spectacular connections between nonassociative algebras and finite groups is the description of the Monster, the largest sporadic finite simple group, as the automorphism group of the Griess algebra. This algebra arises naturally inside a VOA. Recently, some attempts have been made (e.g. by Hall, Rehren and Shpectorov) to axiomatize those algebras, which has led to the notion of axial algebras. In my master thesis, I have introduced modules for axial algebras and I have discovered a connection between the representation theory of Matsuo algebras and the representation theory of 3transposition groups. Our goal is to extend the contemporary theory of axial algebras; this includes a deeper understanding of the finite groups that arise, and a further development of a suitable representation theory.