The lectures will concentrate on quite old,
but also very new investigations concerning the
Brauer-Thrall conjectures, outlining
considerations which complement the usual Auslander-Reiten
methods. In particular, I will focus the attention to the
Gabriel-Roiter measure which was introduced (as "Roiter measure")
by Gabriel in order to make explicit the combinatorial scheme
of Roiter's proof of the first Brauer-Thrall conjecture. This
proof which was intended just for categories of bounded
representation type gives a lot of insight for
artin algebras of arbitrary representation type.