In my lecture I will first present a solution to a question of McKenna and Walter, concerning the positive deformation of hinged plates in a spring bed under positive load. If the spring constant is small, the corresponding differential operator is positivity preserving, but if it gets larger than a critical constant , it is no longer positivity preserving. G.Sweers and I were able to prove that the canonical conjecture is false. Here is a disc of same area as .
In the second part of my lecture I will address the pseudo-Laplace eigenvalue problem, which comes from minimizing on . The Euler-Lagrange equation reads
If there is time left, I will present results on the symmetry of eigenfunctions in situations where
some standard tricks seem to fail.