  We announce a generalization of Zimmer's cocycle superrigidity theorem proven
using harmonic map techniques. This allows us to generalize many results
concerning higher rank lattices to all lattices in semisimple groups with
property $(T)$. In particular, our results apply to SP(1,n) and $F_4^{-20}$ and
lattices in those groups.
  The main technical step is to prove a very general result concerning
existence of harmonic maps into infinite dimensional spaces, namely a class of
simply connected, homogeneous, aspherical Hilbert manifolds. This builds on
previous work of Corlette-Zimmer and Korevaar-Schoen. Our result is new because
we consider more general targets than previous authors and make no assumption
concerning the action. In particular, the target space is not assumed to have
non-positive curvature and in important cases has significant positive
curvature. We also do not make a "reductivity" assumption concerning absence of
fixed points at infinity.
  The proof of cocycle superrigidity given here, unlike previous ergodic
theoretic ones, is effective. The straightening of the cocycle is explicitly a
limit of a heat flow. This explicit construction should yield further
applications.
