In this talk, we will explore three seemingly disparate questions: Do there exist sets on which a nonzero, but finite number of linearly independent harmonic functions can vanish? Are the nodal patterns that arise in Chladni’s famous clamped plated experiment “dense” at the scale of their characteristic wave-length? Is the spectrum of the curl operator on a smoothly bounded domain generically simple? One may shed light on all three of these questions by considering parametric deformations of certain underlying spectral problems.
