The classification of mathematical objects is an important problem in many branches of mathematics, notably also in algebraic geometry. Typically, these classification cannot be achieved by (finite) lists. Instead one builds a new algebraic variety whose points correspond to then isomorphism class of the objects to be classified — a moduli space. This leads to the questions: how does one construct such classifying spaces and what can one say about the geometry of these moduli spaces? I will discuss these question in exemplary cases (old and new).

Classification in algebraic geometry and the geometry of moduli spaces