Abstract The eigenvalues of the fixed membrane problem (corresponding to the eigenvalues of the Dirichlet problem on a domain in Euclidean space) satisfy tight universal and isoperimetric inequalities. In this colloquium, designed to be accessible to grad students, I will survey several problems of interest, and show how the tools of integral transforms can be used in combination with the tools of convex analysis, and the Rayleigh-Ritz principle, to show universal Weyl-sharp inequalities for ratios and averages of these eigenvalues. For wedge-like membranes, and domains in a convex cone, one can in fact improve on the classical Faber-Krahn inequality and prove weighted isoperimetric inequalities which improve on classical results by recourse to the Weinstein interpretation in fractional space, and by introducing weighted forms of rearrangement à la Talenti.