  Pinning particles at random in supercooled liquids is a promising route to
make substantial progress on the glass transition problem. Here we develop a
mean-field theory by studying the equilibrium and non-equilibrium dynamics of
the spherical p-spin model in presence of a fraction c of pinned spins. Our
study shows the existence of two dynamic critical lines: one corresponding to
usual Mode Coupling transitions and the other one to dynamic spinodal
transitions. Quenches in the portion of the c - T phase diagram delimited by
those two lines leads to aging. By extending our results to finite dimensional
systems we predict non-interrupted aging only for quenches on the ideal glass
transition line and two very different types of equilibrium relaxations for
quenches below and above it.
