We return to the problem mentioned above:
We had started with a finite-dimensional hereditary algebra A
and a tilting A-module T. We have denoted by B the endomorphism ring of T.
As we have seen, during the passage from mod A to mod B using the tilting functors
Hom(T,-) and Ext1(T,-), we usually are going to lose some of the A-modules.

Question: Is it possible to recover the lost A-modules, from knowing only B ?

The answer to this question came completely unexpected
- and from outside, from investigations concerning "cluster algebras" as introduced by Fomin and Zelevinsky. There is not enough time to sketch this development,
but we want to outline the final step: the introduction of the "cluster tilted algebras".

Actually: The question was not even asked before the answer was given!