Singularities in Oscillatory Integrals, Inverse Problems and Transformation Optics

本提案涉及四个不同的问题，它们具有统一的主题，即控制、估计或重建在振荡和傅立叶积分算子和反问题中出现的奇点，以及在新兴的变换光学领域中使用奇点来获得新设计的相反目标。第一个问题涉及二维振荡积分算子的最优衰减估计。在第二部分，我们将研究Carleson-Sjolin估计的简并版本。第三个项目涉及地震成像中波动方程的线性化反问题，该问题导致退化傅立叶积分算子的分析和组成。最后，对奇异空间的分析将用于理解隐身和其他变换光学设计的理论限制。这些问题的进展将增加对偏微分方程的理解，用于解决它们的算子，以及它们的解的行为。许多自然定律都用偏微分方程来表示，偏微分方程控制着我们感兴趣的物理量，如电磁场强度、声波施加的压力或粒子在特定位置被发现的量子力学概率。在这个项目中开发的技术适用于控制各种波传播的方程，并且基于理解和操纵存在的奇点的几何观点。特别是，第三个项目有可能在地震震源和接收器之间存在多道射线导致图像失真的情况下促进地震勘探。第四个项目涉及最近兴起的变换光学领域的理论基础，该领域有可能使用被称为超材料的人工结构材料对各种波（例如电磁、声学、量子力学）产生不同寻常的影响。

This proposal deals with four distinct problems having the unifying theme of controlling, estimating or reconstructing singularities that arise in oscillatory and Fourier integral operators and inverse problems, as well as the opposite goal of using singularities to obtain new designs in the emerging field of transformation optics. The first problem concerns optimal decay estimates for oscillatory integral operators in two dimensions. In the second, degenerate versions of the Carleson-Sjolin estimates will be pursued. The third project involves a linearized inverse problem for wave equations arising in seismic imaging which leads to analysis and composition of degenerate Fourier integral operators. Finally, analysis on singular spaces will be used to understand the theoretical limits to cloaking and other transformation optics designs.Progress on these problems will add to the understanding of partial differential equations, the operators which are used to solve them, and the behavior of their solutions. Many laws of nature are expressed as partial differential equations, which govern physical quantities of interest, such as electromagnetic field strength, pressure exerted by acoustic waves or the quantum-mechanical probability that a particle will be found in a particular place. The techniques to be developed in this project are applicable to equations that govern various kinds of wave propagation and are based on a geometric point of view in understanding and manipulating the singularities which are present. In particular, the third project has the potential to facilitate seismic exploration in situations where images are distorted by the presence of multiple rays connecting seismic source to receiver. The fourth project concerns theoretical underpinnings of the recently emergent field of transformation optics, which has the potential to produce unusual effects on various kinds of waves (e.g., electromagnetic, acoustic, quantum mechanical) using artificially structured materials known as metamaterials.