Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$,  and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.

Over the past few decades uniformly rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of singular integrals in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions closely related to the solvability of the Dirichlet problem. In the first lecture will discuss the state of the art in the case of co-dimension one. The second one will concentrate on the new analogue of harmonic measure, recent results, and many remaining mysteries for sets of higher co-dimension