Two central questions of representation theory are the construction of irreducible modules and the construction of larger representations from smaller ones. This project will focus on the latter, which is also called homological algebra. Central questions are: Given an algebra, what extensions do the simple modules have? What is the structure (i.e. composition series) of projective modules? What can we say about the Auslander-Reiten quiver? Its derived category? The PhD project will attack some of these questions, depending on the interests of the candidate for certain algebras appearing in the theory of algebraic groups or categorification, or for abstract  algebras with a given set of homological properties (quasi-heredity, symmetry, cellularity, etc.).

M. Auslander, I. Reiten, S. Smal\o, \emph{Representation theory of Artin algebras}. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.

I. Assem, D. Simson, A. Skowro\'nski, Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge University Press, Cambridge, 2006.