Invariant theoretic approaches to representation theory

Abstract: 

Over the last 40 years there have been a number of attempts to characterize the complexity of the category of modules over a (finite-dimensional associative) algebra by means of invariant theory. More precisely, we seek to determine whether an algebra is tame or wild by considering rings of invariant functions, and the geometry of their moduli spaces. Such characterizations have been demonstrated in the context of path algebras of quivers, but attempts to generalize to bound quiver algebras have met resistance.  I will describe these efforts, their success, their failures, and attempts to address these failures by defining the notion of Schur representation type. All constructions will be illustrated by means of accessible examples.