The Ziegler spectrum of a ring is a topological space, whose points are isoclasses of indecomposable pure injective modules. The concept originally comes from studying modules using model theory (in the sense of mathematical logic), but it can also be nicely interpreted in the language of functor categories.
In case the ring is a finite dimensional algebra over a field, the indecomposable finite dimensional modules are precisely the isolated points of the Ziegler spectrum, but unless the algebra is of finite representation type, there exist infinite dimensional indecomposable pure injective modules as well (i.e. “limit points” in the space). Thus, describing the Ziegler spectrum is a tough problem in general - it is harder than classifying all finite dimensional modules.
The main topic of the talk will be the attempts to classify indecomposable pure-injective modules over tubular canonical algebras. This involves relatively recent results of Angeleri, Kussin and Laking, as well as a joint work with Alessandro Rapa, which is a part of his thesis.