  We study conformal invariants that arise from functions in the nullspace of
conformally covariant differential operators. The invariants include nodal sets
and the topology of nodal domains of eigenfunctions in the kernel of GJMS
operators. We establish that on any manifold of dimension $n\geq 3$, there
exist many metrics for which our invariants are nontrivial. We discuss new
applications to curvature prescription problems.
