U. Haifa Topology & Geometry seminar: Sunday, Jan. 2, 2022. Speaker: Dan Mangoubi (Hebrew University). “Title: A Local version of Courant’s Nodal domain Theorem”

Author: raphy December 30, 2021

Time: Sunday Jan 2nd, 17:00 (Israel Time)

Location: On Zoom

Speaker: Dan Mangoubi (Hebrew University)

Title: A Local version of Courant’s Nodal domain Theorem.

Slides:

Abstract.

Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding to the k-th eigenvalue.

It is easy to see (by direct calculation) that the number of zeros of u_k is exactly k+1.

Equivalently, the number of connected components of the complement of $u_k=0$ is $k$.

In 1923 Courant found that in higher dimensions (considering eigenfunctions of the Laplacian on a closed Riemannian manifold M)

the number of connected components of the open set $M\setminus {u_k=0}$ is at most $k$.

In 1988 Donnelly and Fefferman gave an upper bound on the number of connected components of $B\setminus {u_k=0}$, where $B$ is a ball in $M$.

However, their estimate was not sharp (even for spherical harmonics).

We describe the ideas which give the sharp bound on the number of connected components in a ball.

The talk is based on a joint work with S. Chanillo, A. Logunov and E. Malinnikova, with a contribution due to F. Nazarov.

ZOOM COORDINATES:

Meeting ID: 849 8759 1026