20 oct 2022 -- 14:30   [open in google calendar]

SNS, Aula Mancini

Abstract.

Consider a compact Riemannian manifold with no boundary $(M^n,g)$ where the manifold and metric is smooth. An important object on $M$ is the Laplace-Beltrami operator. Courant's well-known theorem states that if one considers the eigenvalues of this operator in increasing order with multiplicity(we are considering the negative of the Laplacian), then the number of nodal domains of the k-th eigenfunction is at most k. A nodal domain is a connected component of the set where the eigenfunction does not vanish. In our talk we study a problem raised by C. Fefferman and H. Donnelley who raised questions on a local version of Courant's nodal domain theorem. This question is related to a famous conjecture of Yau on the size of the zero set of eigenfunctions. Our work is joint work with A. Logunov, E. Mallinikova and D. Mangoubi.