The class of finite dimensional algebras over an algebraically closed field K
may be divided into two disjoint subclasses (tame and wild dichotomy).
One class
consists of the tame algebras for which the indecomposable modules
occur, in each dimension d, in a finite number of discrete and a
finite number of one-parameter families. The second class is formed by
the wild algebras whose representation theory comprises the
representation theories of all finite dimensional algebras over K.
Hence, the classification of the finite dimensional modules is
feasible only for the tame algebras. Frequently, applying deformations
and covering techniques, we may reduce the study of modules over tame
algebras to that for the corresponding simply connected tame algebras.
We shall discuss the problem concerning connection between the
tameness of simply connected algebras and the weak nonnegativity of
the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.