'INSPIRE Track 1:' Localization: analysis, control, and design of waves in inhomogeneous media

This is an INSPIRE award that is partially funded by the Atomic, Molecular, Optical, and Plasma Physics Program in the Division of Physics, by the Analysis Program in the Division of Mathematical Sciences, and by the Office of Multidisciplinary Activities in the Directorate for Mathematical and Physical Sciences. At the forefront of modern technology, matter structured at the nanometer and atomic levels increasingly reveals its wavelike nature. In recent years, cutting-edge experiments with ultracold atoms, pushing the limits of fundamental physics, have created new states of matter and offered the possibility of controlling quantum entanglement, with major potential applications in cryptography and in quantum computing. The electronic waves confined in quantum wells have yielded high-efficiency, light-emitting diodes that are about to revolutionize the energetics of lighting. At these scales, even the slightest disorder or irregularity can trigger one of the most puzzling and poorly understood phenomena, wave localization. Its dramatic impact has been observed and confirmed by experiments. At the same time, it has been acknowledged that at the present time the design of the most promising light sources (and, more generally, the investigation and exploitation of wave localization) rests on ill-defined concepts, for we still lack the tools necessary to relate disorder to the wave properties in an accurate manner. The present project offers a new approach to wave localization. Its main objective is to combine state-of-the-art techniques and results from harmonic analysis and geometric measure theory with achievements of atomic physics so as to enable one to predict (and then manipulate) localization properties of mechanical, electromagnetic, and matter waves in precise and quantifiable mathematical terms. The project addresses mathematical problems involving the eigenfunctions and behavior of solutions to irregular partial differential equations seamlessly integrated with the latest research in systems of cold atoms, and ultimately aims to give to scientists and engineers a unique opportunity to produce desired wave localization properties on demand, at predetermined frequencies and at specific locations, and at the submillimeter or even atomic scales.

What is wave localization? It is an astonishing ability of physical systems to maintain vibrations in small portions of their original domains of activity, preventing extended propagation. One should not, in this context, think solely in terms of mechanical vibrations. Light is a particular example of an electromagnetic wave, wifi is delivered by waves, sound is a pressure wave, and, from the vantage point of quantum physics, even matter can be perceived as a type of wave. Whether wanted or unwanted, known or ignored, exploited or only sustained, localization of such waves plays a paramount role in our everyday life and will play an even greater one in the science and technology of the future. However, the intricate nature of this phenomenon and the perplexing rules of confinement that govern it remain largely a mystery. The goal of the present project is to establish the right mathematical tools with which to describe and manipulate it, to design and control localization behavior, in contrast to the current situation where researchers have to resort to limited experiments and costly trial-and-error runs. The project's aim is to unveil the fundamental laws of localization and eventually open the way to master vibrations (i.e., redirect, amplify, dampen, focus, and drive them) in all of their manifestations. The results of the research will bring novel methods to bear in analysis, probability, applied mathematics, condensed matter physics, mechanical engineering, acoustics, optics, and quantum physics, to name just a few areas of potential application. In addition, the project will become a platform for a unique interdisciplinary program of training for postdocs and students, with a line-up of activities specifically targeted to involve women and other STEM minorities in scientific research.