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.