  The mode-coupling theory of the glass transition treats the dynamics of
supercooled liquids in terms of two-point density correlation functions. Here
we consider a generalized, hierarchical formulation of schematic mode-coupling
equations in which the full basis of multipoint density correlations is taken
into account. By varying the parameters that control the effective
contributions of higher-order correlations, we show that infinite hierarchies
can give rise to both sharp and avoided glass transitions. Moreover, small
changes in the form of the coefficients result in different scaling behaviors
of the structural relaxation time, providing a means to tune the fragility in
glass-forming materials. This demonstrates that the infinite-order construct of
generalized mode-coupling theory constitutes a powerful and unifying framework
for kinetic theories of the glass transition.
