Explore a comprehensive lecture on the $L^p$ spectrum of the Laplacian on differential forms. Delve into the resolvent set of the Laplacian on $L^p$ integrable $k$-forms, discovering how it lies outside a parabola for manifolds with exponential volume growth rate, without the need for bounded geometry. Examine sufficient conditions for a Weyl criterion to hold on the $L^p$-spectrum of the Laplacian on $k$-forms in open Riemannian manifolds. Gain insights into the detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. Learn about recent findings on the $L^p$ spectrum of conformally compact manifolds. This 59-minute talk by Nelia Charalambous from the University of Cyprus, presented at the Institut des Hautes Etudes Scientifiques (IHES), offers a deep dive into advanced mathematical concepts and recent research results in spectral theory.