  Extending the notion of maximal green sequences to an abelian category, we
characterize the stability functions, as defined by Rudakov, that induce a
maximal green sequence in an abelian length category. Furthermore, we use
$\tau$-tilting theory to give a description of the wall and chamber structure
of any finite dimensional algebra. Finally we introduce the notion of green
paths in the wall and chamber structure of an algebra and show that green paths
serve as geometrical generalization of maximal green sequences in this context.
