Daniel Daners, School of Mathematics and Statistics, University of Sydney Consider all membranes with boundary in the plane which have the same surface area and tension. Lord Rayleigh, in his book "The Theory of Sound," conjectured 1877 that the circular membrane has the lowest ground frequency. The conjecture was proved independently by Faber and Krahn 1923/24 if the membrane is fixed at the boundary. I will look at the corresponding conjecture for the elastically supported membrane, which remained unproved until recently. A partial proof in two dimensions was given by Bossel in 1986. I will outline the ideas for a complete proof for the corresponding problem in arbitrary space dimensions, which settles a recognised old conjecture. This is partly joint work with James Kennedy and Dorin Bucur.