The conformal powers of the Laplacian, known as the GJMS operators, are a family of conformally invariant differential operators with leading term a power of the Laplacian, and the Q-curvature is the associated scalar invariant with conformal properties similar to those of the scalar curvature. They arise naturally and provide a unifying framework connecting important developments in geometry, analysis, and mathematical physics.

In this talk, I will consider the question of the compactness of constant Q-curvature metrics on a closed Riemannian manifold. I will outline the background, motivations, and key challenges, and present some of my results (with collaborators) in this direction.