Singularities in Oscillatory Integrals, Inverse Problems and Transformation Optics

这个建议涉及四个不同的问题，具有统一的主题，控制，估计或重建奇点，出现在振荡和傅立叶积分算子和逆问题，以及相反的目标，使用奇点，以获得新的设计在新兴领域的变换光学。第一个问题是二维振荡积分算子的最优衰减估计。在第二个，退化版本的Carleson-Sjolin估计将继续进行。第三个项目涉及地震成像中产生的波动方程的线性化反问题，导致退化傅立叶积分算子的分析和合成。最后，奇异空间的分析将被用来理解隐形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，用于解决这些问题的算子，以及它们的解决方案的行为。许多自然定律都被表达为偏微分方程，这些方程支配着感兴趣的物理量，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各种波传播的方程，并以理解和操纵存在的奇点的几何观点为基础。特别是，第三个项目有可能在图像因连接震源和接收器的多条射线而失真的情况下促进地震勘探。第四个项目涉及最近出现的变换光学领域的理论基础，该领域有可能对各种波产生不寻常的影响（例如，电磁的、声学的、量子力学的），使用被称为超材料的人工结构材料。

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.