Topics in Wave Propagation and Quantum Systems

波动现象在自然界中无处不在，几乎存在于日常生活的方方面面。它们负责我们听到的声音，看到的光，以及我们在个人设备上接收到的数据。因此，了解如何对这些波进行数学建模，如何控制它们，以及如何从中提取有价值的信息是各个领域交叉的一个重要问题。波传播背后的基本物理原理极其复杂，数学已被证明是他们研究的关键工具。这个项目的核心是对波动现象的理论分析和模拟，重点关注三个主要问题：如何用最小的实验设备集中能量；如何利用波传播推断复杂介质的物理性质以及如何在新型量子设备中模拟波。该项目将研究与培养应用数学领域的下一代科学家结合起来。本项目的研究主题是波动现象，包括经典波和量子波。拟议的研究有三个主要组成部分。第一个源于最近对瞬时时间镜的突破性实验。后者为时间反转和波的控制提供了新的途径，是数学家们探索的一个令人兴奋的新课题。第二个部分涉及基于随机校正器的逆问题，它被认为是物理系统，其中正向数据是具有小参数和随机系数的偏微分方程的解。在这些系统中，数据由随机校正器渐近建模到一个无法测量的主要项。目标首先是表征这些随机校正，其次是开发基于这些校正的反演策略来提取系统信息。一个典型的例子是“海冰问题”，它涉及均匀化解周围的随机波动。这个项目的最后一个组成部分涉及量子系统。第一个问题是建立基于熵原理的量子流体力学模型的数学基础，第二个问题是推导周期狄拉克算子的有效数值格式，并将其应用于物质的奇异相研究和新型量子器件的设计。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Wave phenomena are ubiquitous in nature, and are virtually in every aspect of every day life. They are responsible for the sound we hear, the light we see, and the data we receive on our personal devices. Understanding how to mathematically model these waves, how to control them, and how to extract valuable information from them is therefore an important question at the intersection of various fields. The underlying physics principles behind wave propagation are extremely complex, and mathematics has proven to be a crucial tool in their investigation. At the core of this project are the theoretical analysis and the simulation of wave phenomena, with an emphasis on three main problems: how to focus energy with a minimal experimental apparatus; how to infer physical properties of a complex medium using wave propagation; and how to model waves in novel quantum devices. This project integrates research with the training of the next generation of scientists in the field of applied mathematics.The main theme of research of the project is wave phenomena, including both classical and quantum waves. There are three main components to the proposed research. The first one originates from the recent groundbreaking experiments on instantaneous time mirrors. The latter form a novel avenue for time reversal and the control of waves, and is an exciting new topic for mathematicians to explore. The second component concerns inverse problems based on stochastic correctors, which are considered physical systems where forward data are solutions to PDEs with small parameters and random coefficients. In these systems, the data are asymptotically modeled by random correctors to a leading term that is not accessible to measurements. The objectives are first to characterize these stochastic correctors, and second to develop inversion strategies based on these corrections to extract information on the system. A typical example is the “sea ice problem”, which involves random fluctuations around a homogenized solution. The last component of this project concerns quantum systems. The first problem is to develop the mathematical foundations of quantum hydrodynamical models based on the entropy principle, and the second the derivation of efficient numerical schemes for periodic Dirac operators, with applications to the study of exotic phases of matter and the design of novel quantum devices.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.