Explore the critical points of Steklov eigenfunctions in this 46-minute seminar talk by Luigi Provenzano from Università di Roma. Delve into the study of compact, smooth n-dimensional Riemannian manifolds with boundaries and discover an identity relating the sum of indexes of Steklov eigenfunctions, their boundary restrictions, and the Euler characteristic for generic metrics. Learn how this identity provides a precise count of interior critical points in two-dimensional cases, connecting the Euler characteristic and sign changes on the boundary. Gain insights from Provenzano's joint work with Luca Battaglia and Angela Pistoia, presented as part of the Spectral Geometry in the clouds seminar series at the Centre de recherches mathématiques.