Upper bounds on the size of nodal sets of eigenfunctions

Abstract: This talk discusses a problem introduced by Yau on estimating the size of nodal sets of eigenfunction in terms of the eigenvalue. We show that one can obtain improved polynomial upper bounds when the Riemannian manifold has a Gevrey or quasianalytic regularity. Yau's upper bound conjecture was proved by Donnelly and Fefferman in the analytic case. Logunov proved a polynomial upper bound in terms of the eigenvalue in the smooth case. We will be concerned with intermediate classes of Riemannian manifolds.