Counting representations of algebras

How could one understand an algebraic object, for example, a group or, more generally, an algebra? One efficient way is to look at how it interacts with well-understood, simpler mathematical structures, for example, vector spaces. This is the subject of representation theory. If one tries to understand the properties of an algebraic variety, that is, the set of common solutions of some polynomial equations, one may count the solutions over finite fields. This is part of algebraic and arithmetic geometry. Following an idea of Kac, we bring these two problems together and aim at understanding the geometry of some spaces arising from the representations of a given algebra. More precisely, we count the representations of an algebra up to isomorphism and obtain a surprising polynomial behaviour (under explicit assumptions). The group algebra of the modular group fits under this umbrella, pointing at possible consequences in arithmetic or in geometric group theory. The versions of these polynomials for quivers have fostered huge developments in geometric representation theory since the 1980s, and I will explain how this story can be extended to smooth algebras. This is all joint work with Fabian Korthauer (Düsseldorf).