  The AdS/CFT correspondence allows us to map a dynamical cosmology to a dual
quantum field theory living on the boundary of spacetime. Specifically, we
study a five-dimensional model cosmology in type IIB supergravity, where the
dual theory is an unstable deformation of $\N=4$ supersymmetric SU(N) gauge
theory on $\Rbar\times S^3$. A one-loop computation shows that the coupling
governing the instability is asymptotically free, so quantum corrections cannot
turn the potential around. The big crunch singularity in the bulk occurs when a
boundary scalar field runs to infinity, in finite time. Consistent quantum
evolution requires that we impose boundary conditions at infinite scalar field,
i.e. a self-adjoint extension of the system. We find that quantum spreading of
the homogeneous mode of the boundary scalar leads to a natural UV cutoff in
particle production as the wavefunction for the homogeneous mode bounces back
from infinity. However a perturbative calculation indicates that despite this,
the logarithmic running of the boundary coupling governing the instability
generally leads to significant particle production across the bounce. This
prevents the wave packet of the homogeneous boundary scalar to return close to
its initial form. Translating back to the bulk theory, we conclude that a
quantum transition from a big crunch to a big bang is an improbable outcome of
cosmological evolution in this class of five-dimensional models.
