On the occasion of Robert Osserman’s retirement

1272 NOTICES OF THE AMS VOLUME 42, NUMBER 11 I n the late 1970s, when I was a student at Stanford University, I attended a seminar run by Bob Osserman on eigenvalue problems—more specifically—bounds for eigenvalues. This seminar was instructive for me and my fellow students. It kindled my keen interest in spectral geometry. One of the results reported in this seminar was due to Osserman; I would like to review it as a starting point for this lecture. Suppose W is a smooth, simply connected domain in R2. We are interested in the smallest positive eigenvalue for the Laplacian ∆ on W with Dirichlet boundary conditions: λDIR(W ). In other words, the bass note of the drum W :