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 −20 4 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.

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Abstract. In this paper, we propose a property which is a natural generalization of Kazhdan's property (T ) and prove that many, but not all, groups with property (T ) also have this property.

In this editorial, we provide an introduction to the special issue on Inquiry-Based Learning in First and Second Year Courses. We also discuss the essential features of inquiry-based learning and provide a brief overview of the literature and evidence for its effectiveness.

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