  This work studies front formation in the Allen-Cahn equation with a parameter
heterogeneity which slowly varies in space. In particular, we consider a
heterogeneity which mediates the local stability of the zero state and
subsequent pitchfork bifurcation to a non-trivial state. For slowly-varying
ramps which are either rigidly propagating in time or stationary, we rigorously
establish existence and stability of positive, monotone fronts and give leading
order expansions for their interface location. For non-zero ramp speeds, and
sufficiently small ramp slopes, the front location is determined by the local
transition between convective and absolute instability of the base state and
leads to an O(1) delay beyond the instantaneous pitchfork location before the
system jumps to a nontrivial state. The slow ramp induces a further delay of
the interface controlled by a slow-passage through a fold of strong- and
weak-stable eigenspaces of the associated linearization. We introduce
projective coordinates to de-singularize the dynamics near the trivial state
and track relevant invariant manifolds all the way to the fold point. We then
use geometric singular perturbation theory and blow-up techniques to locate the
desired intersection of invariant manifolds. For stationary ramps, the front is
governed by the slow passage through the instantaneous pitchfork bifurcation
with inner expansion given by the unique Hastings-McLeod connecting solution of
Painlev\'{e}'s second equation. We once again use geometric singular
perturbation theory and blow-up to track invariant manifolds into a
neighborhood of the non-hyperbolic point where the ramp passes through zero and
to locate intersections.
