Abstract:

Let (M,g) be a compact closed Riemannian manifold. Let f_k be a sequence of eigenfunctions of the Laplace-Beltrami operator on M with increasing eigenvalues \lambda_k. When one looks at the set {f_k=0} as k goes to infinity, one discovers beautiful patterns, the study of which was already initiated by Chladni in the 17th century. In the last 100 years and in particular in the last 25 years there have been several interesting results in the subject. In particular, the connection to properties of harmonic functions in Euclidean space was understood. We will describe those achievements and also a few quite challenging open questions.