The examples will be: [1] Spectral determinants of conformally covariant operators (e.g. conformal Laplace and Dirac operators), and [2] Total $Q$-curvature. Both of these satisfy conformal invariance. We recently found a striking universality in the variational structure when the base manifold is the round sphere, in any such problem, by using representation theory of the conformal group. As a corollary it gives us a neat proof of some of the extremal results in Kate Okikiolu's paper from Annals (2001), and of analogous results for many other examples. Lastly, I will mention the proof of $B\% \ r-$ Schopka's conjecture on dimensional asymptotics of explicit determinants on spheres (i.e. the extremal values in the examples [1] above). Some of the work is joint with Bent Orsted.