Representation type and varieties

More than 50 years ago, Higman found a precise connection between the subgroup structure of a finite group and its number of representations, that is vector spaces on which it can act. Bondarenko and Drozd refined Higman's work some time later. More generally, finite dimensional algebras come in three flavors, or representation types: finite, tame, and wild. In this introductory talk, we will give an overview of representation type and of support varieties, which are useful invariants of representations. We will explain how support varieties have been used recently to obtain wildness criteria for some algebras. Specific examples include some Hopf algebras, small quantum groups, and related algebras.