Microlocal analysis for waves and inverse problems

计划中的研究开发和应用微局部分析领域的工具。粗略地说，这个场同时跟踪波(或更广泛地说，函数)的位置和频率或动量。计划的应用是波传播和其他相关现象，以及根据曲线积分确定函数的反问题(X射线变换)，以及根据边界测量确定材料结构的相关问题。尽管该提议涉及他们的数学理论，但这些问题与物理世界密切相关。波的传播在自然界是无处不在的，电磁波，如光，是最普遍的例子之一；广义相对论是另一个重要的物理例子。量子粒子(如质子和电子)的散射理论是微局域分析的另一个主题：这些方面既涉及到对大距离量子波的描述，也涉及到半经典现象，即当普朗克常数可以被认为是小的时，这在化学中经常发生。正在研究的反问题也具有广泛的意义：这里开发的理论的一个应用是通过测量波的传播时间来确定物体中未知的可变声速，例如，这与利用地震波的传播时间成像到地球内部有关。一些拟议的项目描述了波在弯曲时空上的长时间或远场行为。从物理上讲，这些都是广义相对论产生的，包括弯曲背景上的电磁波。然而，也有纯数学起源的例子，如渐近复双曲(ACH)空间和渐近反德西特空间(ADS)；后者通过弦理论与物理学有不同的联系。在这些空间上的微局部分析方法使得作者在渐近(实)双曲(AH)空间以及Kerr-de Sitter空间上线性问题的工作取得了突破。本文的目的是将这些工具推广到ACH，ADS-型空间，提高对洛伦兹散射空间(包括渐近Minkowski空间)以及拟线性偏微分方程解的理解。其他项目涉及波在边缘的行为，具体地说是弹性瑞利(表面)波的绕射，以及标量波或电磁波绕射的精细的第二微局部描述。另一个主要领域是反问题，作者与乌尔曼一起介绍了测地线X射线变换的空间局域反演的新工具。这些项目的目的是将其推广到张量，并研究边界刚性，即从边界距离函数恢复黎曼度量。

The planned research develops and applies tools of the field of microlocal analysis. Roughly speaking, this field keeps track of the position and frequency, or momentum, of waves (or more generally, functions) simultaneously. The planned applications are to wave propagation and other related phenomena, as well as inverse problems for determining a function from integrals along curves (X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the proposal concerns their mathematical theory, these problems are closely connected to the physical world. Wave propagation is ubiquitous in nature, with electromagnetic waves, such as light, being one of the most prevalent examples; the theory of general relativity being another important physical example. Scattering theory of quantum particles (such as protons and electrons) is another subject governed by microlocal analysis: these aspects enter both into the description of quantum waves at large distances, and into semiclassical phenomena, i.e. when Planck's constant can be regarded as small, which happens often in chemistry. The inverse problems under study are also of broad significance: an application of the theory developed here is the determination of an unknown variable sound speed in an object via the measurement of travel times of waves, which for instance is relevant to imaging to interior of Earth using the travel times of earthquake waves.Some of the proposed projects describe the long-time or far field behavior of waves on curved space-times. Physically these arise in general relativity, including electromagnetic waves on a curved background. However, there are also examples of purely mathematical origin, such as asymptotically complex hyperbolic (ACH) space and asymptotically Anti de Sitter spaces (AdS); the latter does have a different connection with physics via string theory. The microlocal approach to analysis on these spaces has made breakthroughs possible in the author's work on linear problems asymptotically (real) hyperbolic (AH) spaces as well as Kerr-de Sitter space. The projects here aim to extend these tools to the ACH, AdS-type spaces, improve the understanding of Lorentzian scattering spaces (which include asymptotically Minkowski spaces), as well as to quasilinear PDE. Other projects concern the behavior of waves at edges, concretely the diffraction of the Rayleigh (surface) waves of elasticity and a refined, second-microlocal, description of diffraction of scalar or electromagnetic waves. Yet another main area is inverse problems, where the author, together with Uhlmann, has introduced new tools for spatially localized inversion of the geodesic X-ray transform. The projects aim to extend this to tensors, and to investigate boundary rigidity, i.e. recovering a Riemannian metric from its boundary distance function.