Nodal sets are zero sets of eigenfunctions of the Laplacian on a Riemannian manifold. Local analysis studies nodal sets in small balls, ignoring the global geometry. Global analysis exploits the dynamics of the geodesic flow to obtain information on nodal sets. First, I will describe the recent proof by Alexander Logunov of Yau’s lower bound conjecture for hypersurface volumes of nodal sets. It is a local proof based mainly on the combinatorics of the Donnelly-Fefferman doubling exponent bounds. Second, I will describe recent results on numbers of nodal domains on surfaces of non-positive curvature. These results are based on the ergodicity of the geodesic flow (joint work in part with Junehyuk Jung, related work by Ghosh-Reznikov-Sarnak).
This video is part of Harvard University‘s conference JDG 2017.
