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

这是一项INSPIRE奖，部分资金来自物理部的原子、分子、光学和等离子体物理计划，数学科学部的分析计划，以及数学和物理科学局的多学科活动办公室。在现代技术的前沿，在纳米和原子水平上结构的物质越来越显示出它的波动性质。近年来，用超冷原子进行的尖端实验突破了基础物理的极限，创造了新的物质状态，并提供了控制量子纠缠的可能性，在密码学和量子计算中具有重要的潜在应用。限制在量子井中的电子波产生了高效率的发光二极管，即将给照明的能量学带来革命性的变化。在这些尺度上，即使是最轻微的无序或不规则性也可能引发最令人费解和鲜为人知的现象之一--波的局部化。它的巨大影响已经被实验观察和证实。与此同时，人们已经认识到，目前最有希望的光源(以及更广泛地说，波局部化的研究和开发)的设计依赖于定义不明确的概念，因为我们仍然缺乏必要的工具来准确地将无序与波的性质联系起来。本课题为波的定位提供了一种新的途径。它的主要目标是将调和分析和几何测量理论的最新技术和结果与原子物理学的成果相结合，使人们能够用精确和可量化的数学术语预测(然后操纵)机械波、电磁波和物质波的局域化性质。该项目解决了涉及不规则偏微分方程解的本征函数和行为的数学问题，与冷原子系统的最新研究无缝地结合在一起，最终旨在为科学家和工程师提供一个独特的机会，根据需要在预定频率和特定位置，甚至在亚毫米甚至原子尺度上产生所需的波局部化特性。什么是浪潮本地化？这是一种惊人的能力，物理系统能够将振动保持在其原始活动区域的一小部分中，防止扩展传播。在这种情况下，人们不应该只从机械振动的角度来思考。光是电磁波的一个特殊例子，wifi是通过波传递的，声音是一种压力波，从量子物理的角度来看，即使是物质也可以被认为是一种波。无论是想要的还是不想要的，知道的还是忽视的，被利用的或仅仅是持续的，这种浪潮的本地化在我们的日常生活中发挥着至关重要的作用，并将在未来的科学技术中发挥更大的作用。然而，这一现象的错综复杂的本质以及管理它的令人费解的限制规则在很大程度上仍然是一个谜。本项目的目标是建立正确的数学工具来描述和操纵它，设计和控制本地化行为，而不是目前研究人员不得不求助于有限的实验和代价高昂的反复试验的情况。该项目的目标是揭示本地化的基本规律，并最终为掌握所有表现形式的振动(即重定向、放大、抑制、聚焦和驱动)开辟道路。研究结果将在分析、概率、应用数学、凝聚态物理、机械工程、声学、光学和量子物理等领域带来新的方法，仅举几个潜在的应用领域。此外，该项目将成为一个平台，为博士后和学生提供独特的跨学科培训计划，并推出一系列专门针对妇女和其他STEM少数民族参与科学研究的活动。

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.