The first lecture from the minicourse "Courant, Bezout, and Persistence taught by Leonid Polterovich (Tel Aviv University)

I'll discuss an approach to study oscillation of functions based on persistence modules and barcodes, an area of algebraic topology
originating in data analysis. This technique has a number of applications to spectral geometry, a discipline dealing, in particular, with geometric features of eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds. The first one is an extension of Courant's nodal domain theorem to linear combinations of eigenfunctions, a problem with long history, with a caveat being that the direct generalization is known to be false. The second application is a version of Bezout's theorem in the context of eigenfunctions, supporting an intuition, due to H.Donnelly and C.Fefferman, that eigenfunctions behave as polynomials whose degree is comparable to the square root of the eigenvalue. The necessary preliminaries from topological persistence and spectral geometry will be explained.

The lectures are based on a paper "Coarse nodal count and topological persistence" with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin, and Vukašin Stojisavljević, arXiv:2206.06347.