Day 1 Session 6 by Sagun Chanillo

Title: Local Version of Courant’s Nodal Domain Theorem

Abstract:
Consider a smooth and compact Riemannian manifold (Mn, g) with no boundary and equipped with a smooth metric g. The Laplace-Beltrami operator Δg is an object on M that has been the subject of much study with early work by H. Weyl and Minakshisundaram-Pleijel. Consider the eigenvalues λk and associated eigenfunctions uk of Δg given by
−Δguk = λkuk.

[sanjayshastri]

It seems the speaker and his collaborators have solved a 30 year old conjecture in the Geometry of the Laplace-Beltrami operator on compact Riemannian manifolds.