A brief introduction to the representation theory of quivers and to cluster-tilting theory

The main objective of representation theory is the study of algebras via the study of their modules (i.e. representations). In this mini-course we will consider a combinatorial approach to this theory, using quivers (i.e. oriented graphs) and their representations (given by vector spaces associated to vertices and linear maps associated to arrows). This approach can be used to study the representation theory of all finite-dimensional algebras, since these can be realised as quotients of path algebras of a quiver.