Localization of vibrations is one of the most intriguing features exhibited by irregular or inhomogeneous media. A striking (but certainly not unique) example is the so called 'Anderson localization' of quantum states by a random potential, that was discovered by Anderson in 1958 and that brought him the Nobel Prize in 1977. Anderson localization is one of the central topics in condensed matter physics, producing hundreds of papers each year. Yet, there exists up to now no theoretical framework able to predict exactly what triggers localization, where it happens and at which frequency. We present a fundamentally new mathematical approach that explains how the system geometry and the differential operator intervening in the wave equation interplay to give rise to a ``landscape" that reveals weakly coupled subregions inside the system, and how these regions shape the spatial distribution of the vibrational eigenmodes. This is joint work with Marcel Filoche. Svitlana Mayboroda is an Associate Professor at the School of Mathematics at the University of Minnesota. Her research interests include elliptic partial differential equations, harmonic analysis, and geometric measure theory. She graduated from the University of Missouri in 2005 and held positions at the Ohio State University, Brown University, Australian National University, and Purdue University before coming to Minnesota. Svitlana Mayboroda has been a recipient of the Alfred P. Sloan Fellowship and NSF CAREER award. She is the creator of the Workshop for Women in Analysis and PDEs.