Representation theory of algebras is a rapidly evolving area of research with many interrelations to other parts of mathematics. A striking number of new techniques have been developed in the last 25 years which provide the possibilities for a detailed insight into the structure of a module category and which yield algorithms in order to use computers for actual calculations. The geometric invariant theory approach to representation theory leads to the consideration of quadratic forms on Grothendieck groups; these quadratic forms often can be recovered homologically.
The devices of homological algebra were successfully used during the last three decades for establishing the basic notions of representation theory, starting with the work of Auslander and Reiten on almost split sequences. The Auslander-Reiten quivers obtained in this way provide a a good combinatorial model for the behavior of the whole module category and allow a deeper insight in the structure of the module category. The tilting theory relates algebras with not necessary equivalent module categories.
The reduction of problems in representation theory to discrete problems allows the use of computers. Computational methods play already now an important role: on the one hand, they provide an effective tool to search for counterexamples, on the other hand, the combinatorial approach usually leads to the consideration of a large number of exceptional cases which only can be handled by computers. Several powerful algorithms dealing with representations of finite groups and later also for algebras have been available for some time; computer algebra systems have been used very successfully to solve several problems and the design and implementation of algorithms will continue to constitute a major topic of interest.
The four Euroconferences organized by the Network "Invariants and Representations of Algebras".