Graph Structure of the Nodal Set on Riemannian Manifolds

Explore a 40-minute mathematical seminar presentation examining the intricate relationships between eigenfunction zero sets, graph theory, vanishing order of eigenfunctions, and unique continuation in Riemannian manifolds. Delve into the analysis of nodal sets of eigenfunctions within the Schrödinger operator framework on compact, orientable Riemannian manifolds, where they are identified as imbedded metric graphs. Learn how elementary graph theory techniques are applied to estimate both the number of critical points in the nodal set of the k-th eigenfunction and the sum of vanishing orders at critical points, expressed in terms of k and the manifold's genus. The presentation, part of the Spectral Geometry in the Clouds seminar series, draws from collaborative research with Matthias Täufer at Fern Universität in Hagen.