Singularities in Oscillatory Integrals and Inverse Problems

该项目涉及几个不同的问题，它们的共同主题是控制、估计或重建振荡积分算子、傅立叶积分算子和反问题中出现的奇点。第一个问题是建立具有一般齐次多项式相位函数的三维振荡积分算子的最优衰减率。在第二章中，我们将分析二维具有跳跃不连续的标量电导率的Calderon问题，边界数据的奇异性(适当地解释)产生关于内部电导率跳跃的位置和可能大小的信息。在第三个问题中，将使用微局部分析和调和分析的技术来分析衰减Radon变换识别问题的线性化版本。最后，线性化的双曲反问题涉及到研究与特殊奇异几何有关的傅立叶积分算子的组成，这些算子是由三维中普遍存在的焦散引起的。在这些问题上的进展将增加对偏微分方程组、用于求解它们的算子以及它们的解的行为的理解。自然界的许多定律都用偏微分方程式表示，这些偏微分方程式支配着感兴趣的物理量，例如电磁场强度或声波施加的压力。这个项目中将要开发的技术适用于控制各种波传播的方程，并基于几何观点来理解这些奇异性，这些奇异性是存在的。后三个问题有可能有助于无损检测、医学成像和地震勘探系统的设计或改进。

ABSTRACT The project involves work on several distinct problems with the commonthemes of controlling, estimating or reconstructing singularities that arise inoscillatory integral operators, Fourier integral operators and inverse problems.The first problem concerns establishing optimal decay rates for oscillatoryintegral operators in three dimensions having generic homogeneous polynomialphase functions. In the second, the Calderon problem for scalar conductivitieswith jump discontinuities in two dimensions will be analyzed, with singularities(suitably interpreted) of boundary data yielding information about thelocations, and possibly sizes, of the jumps of the conductivity in the interior. A linearized version of the identification problem for the attenuated Radontransform will be analyzed using techniques from micro-local and harmonic analysisin the third problem. Finally, a linearized hyperbolic inverse problem involvesstudying the composition of Fourier integral operators associated to specificsingular geometries arising from the presence of caustics occurring genericallyin three dimensions.Progress on these problems will add to the understanding of partial differentialequations, the operators, which are used to solve them, and thebehavior of their solutions. Many laws of nature are expressed as partial differentialequations, which govern physical quantities of interest, such as electromagneticfield strength or pressure exerted by acoustic waves. The techniques to be developedin this project are applicable to equations that govern various kinds of wavepropagation and are based on a geometric point of view in understanding thesingularities, which are present. The last three problems have the potential toassist in the design or refinement of systems for nondestructive testing, medical imaging, and seismic exploration.