Trees, Representations and Exotic Fusion Systems

Saturated fusion systems are both a convenient language in which to formulate p-local finite simple group theory and interesting structures in their own right. In this talk, we will start by explaining what is meant by a 'tree of fusion systems' and give conditions on such an object for there to exist a saturated completion. We then describe how this theory can be used to understand a class of fusion systems first considered by Bob Oliver, which are determined by modular representations of finite groups. If time permits, we will discuss joint work with David Craven towards a complete classification of such fusion systems. The talk is aimed at a general mathematical audience with some background in algebra.