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

这是一个INSPIRE奖，部分由物理系的原子，分子，光学和等离子体物理计划，数学科学系的分析计划以及数学和物理科学局的多学科活动办公室资助。在现代技术的最前沿，在纳米和原子水平上结构的物质越来越多地显示出其波动性质。近年来，超冷原子的尖端实验突破了基础物理学的极限，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。限制在量子威尔斯井中的电子波已经产生了高效率的发光二极管，它即将彻底改变照明的能量学。在这些尺度上，即使是最轻微的混乱或不规则也会引发最令人困惑和最不了解的现象之一，波局部化。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，目前最有前途的光源的设计（以及更普遍地说，波局部化的研究和利用）依赖于定义不清的概念，因为我们仍然缺乏以准确的方式将无序与波的性质联系起来的必要工具。本项目提供了一种新的方法来波定位。它的主要目标是联合收割机国家的最先进的技术和结果，从谐波分析和几何测量理论与原子物理学的成就，使人们能够预测（然后操纵）的本地化性能的机械，电磁和物质波在精确和可量化的数学术语。该项目解决的数学问题涉及本征函数和不规则偏微分方程的解的行为，与冷原子系统的最新研究无缝集成，最终旨在为科学家和工程师提供一个独特的机会，在预定频率和特定位置，以及亚毫米甚至原子尺度上按需产生所需的波定位特性。什么是wave localization？这是物理系统的一种惊人的能力，能够在它们原始活动领域的一小部分中维持振动，防止扩展传播。在这种情况下，人们不应该仅仅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，从量子物理学的Vantage来看，甚至物质也可以被视为一种波。无论是想要的还是不想要的，已知的还是被忽视的，利用的还是持续的，这种波的定位在我们的日常生活中起着至关重要的作用，并将在未来的科学和技术中发挥更大的作用。然而，这一现象的复杂性质和管理它的令人困惑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，与研究人员不得不诉诸有限的实验和昂贵的试错运行的现状相反。该项目的目的是揭示定位的基本规律，并最终开辟掌握振动的道路（即，重定向、放大、抑制、聚焦和驱动它们）。研究结果将带来新的方法，以承担分析，概率，应用数学，凝聚态物理，机械工程，声学，光学和量子物理，仅举几个潜在的应用领域。此外，该项目将成为博士后和学生独特的跨学科培训计划的平台，并将开展一系列专门针对妇女和其他STEM少数群体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.