In this talk we will study the generalized polynomial chaos (gPC) approach to
with uncertain coefficients/inputs, and multiple time or space scales, and
show that they can be made
asymptotic-preserving or well-balanced, in the sense
that the gPC scheme preserves various asymptotic limits in the discrete space.
This allows the implemention
of the gPC methods for these problems without numerically resolving (by space,
time, and gPC modes) the small scales. We also give a fast gPC algorithm for
the Boltzmann equation
with uncertainties in its collision kernel, initial or boundary data.