Courant, Bezout, and Persistence in Spectral Geometry - Lecture 1

Explore the first lecture of a minicourse on "Courant, Bezout, and Persistence" delivered by Leonid Polterovich from Tel Aviv University. Delve into an innovative approach to studying function oscillation using persistence modules and barcodes, a technique rooted in algebraic topology and data analysis. Discover how this method applies to spectral geometry, particularly in analyzing eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds. Examine an extension of Courant's nodal domain theorem for linear combinations of eigenfunctions, addressing a long-standing problem in the field. Investigate a version of Bezout's theorem in the context of eigenfunctions, supporting Donnelly and Fefferman's intuition about eigenfunction behavior. Gain essential background knowledge in topological persistence and spectral geometry throughout this 58-minute lecture, which draws from the paper "Coarse nodal count and topological persistence" co-authored by Polterovich and colleagues.