Modular representation theory studies representations of a finite group over a field of positive characteristic. Except in a handful of cases, it is impossible to classify modular representations, making the theory “wild,” even for a group of size 8! Modular representation theory, more broadly construed, includes the representation theory of numerous other algebraic objects, such as positive characteristic Lie algebras.

Associating geometric invariants to modular representations allows one to give some structure to this wild territory and even parameterize naturally occurring classes of representations. We’ll start with Quillen’s classical work on group cohomology and move to more recent developments using small examples of finite groups for visual illustration of the general theory.