Oscillatory Integral Operators, Inverse Problems and Non-Transformation Optics

这个建议包括四组问题。其中三个涉及几何和分析之间的相互作用，函数的研究和算子的性质，这是将函数转换为新函数的变换。在这个项目中研究的运营商出现时，试图了解波如何传播（例如，声波在地球上），或在用于研究波传播的基本数学工具，或在研究的几何性质的大集合的点。波可以被理想化为沿着沿着传播，例如光线，当光线集中在一个小区域时，需要开发新的方法来获得波的精确描述。该项目的两个部分将继续主要研究者对各种算子的研究，以及如何从射线结构或更一般的基本几何结构预测它们的属性。可能的应用包括提高对地震成像中伪影的理解。第三部分涉及多线性算子，它一次作用于多个函数。这些问题出现在研究两个或多个波如何相互作用以及离散几何中点云的研究中。第四部分是一种新的方法来设计谐振设备，如天线。材料科学、物理学和数学的最新进展导致了由结构化复合材料（也称为超材料）构造的器件的设计的快速进步，所述结构化复合材料对波的传播具有根本性影响。一种特别成功的方法是基于变换光学。首席研究员将研究一种新的非变换光学设计方法。两个拟议的项目涉及开发新的工具，用于处理退化或奇异的光滑或实解析函数。在一个，退化傅里叶积分算子与光滑相位的组合导致算子具有不光滑的波前关系，并且理解如何将傅里叶积分算子类算子与这些几何相关联将扩展微局部分析的范围并帮助分析某些逆问题。在另一方面，需要找到新的技术来处理振荡积分算子的实分析阶段，在一加二或二加二维。困难包括理解在多大程度上两个或多个函数可以同时解决它们的零变种。一些相同的技术也将被应用于试图找到跳跃数的算法描述，这是奇点理论和代数几何中当前感兴趣的主题。这样做将加强分析与这些领域之间的联系。 该项目第三部分的进展不仅将进一步发展调和分析中的多线性算子理论，而且还将直接应用于几何测度理论和离散几何。 最后一部分将展示如何建立从线性偏微分方程的想法可以在设计和严格验证谐振结构，如天线的属性是有用的，并将有助于促进一个新的设计方法的快速发展的超材料领域。

This proposal consists of four groups of problems. Three of them are concerned with the interaction between geometry and analysis, the study of functions and the properties of operators, which are transformations that turn functions into new functions. The operators studied in this project arise either when trying to understand how waves propagate (for example, acoustic waves in the Earth), or in basic mathematical tools that are used to study wave propagation, or in the study of geometric properties of large sets of points. Waves can be idealized as traveling along rays, such as light rays, and when the rays concentrate in a small region, new methods need to be developed to obtain accurate descriptions of the waves. Two parts of this project will continue the principal investigator's study of various operators and how their properties can be predicted from the structure of rays or more general underlying geometry. Possible applications include improved understanding of artifacts in seismic imaging. A third part concerns multilinear operators, which act on several functions at a time. These arise in the study of how two or more waves interact and also in the study of point clouds in discrete geometry. The fourth part is a new approach to the design of resonant devices, such as antennas. Recent progress in materials science, physics, and mathematics has led to rapid advances in the design of devices constructed from structured composite materials, also called metamaterials, which have radical effects on wave propagation. One particularly successful approach is based on transformation optics. The principal investigator will investigate a new, non-transformation optics design methodology. Two of the proposed projects concern developing new tools for dealing with degeneracies or singularities of smooth or real-analytic functions. In one, composition of degenerate Fourier integral operators with smooth phases leads to operators that have wave-front relations that are not smooth, and understanding how to associate Fourier-integral-operator-like operators to these geometries will expand the reach of microlocal analysis and help analyze certain inverse problems. In another, new techniques will need to be found to deal with oscillatory integral operators with real-analytic phases in one plus two or two plus two dimensions. The difficulties include understanding to what extent two or more functions can have their zero varieties simultaneously resolved. Some of the same techniques will also be applied to try to find an algorithmic description of jumping numbers, which are the subject of current interest in singularity theory and algebraic geometry. Doing so will strengthen the connection between analysis and these fields. Progress on the third part of the project will not only further develop the theory of multilinear operators within harmonic analysis, but it will also have immediate applications to geometric measure theory and discrete geometry. The final component will show how established ideas from linear partial differential equations can be useful in designing and rigorously verifying the properties of resonant structures such as antennas and will help contribute a new design methodology to the rapidly developing area of metamaterials.