Microlocal Analysis of Linear and Nonlinear Problems

该项目开发并应用了被称为微局部分析的数学领域工具。粗略地说，微局部分析设计了同时跟踪波（或更一般地说，函数）的位置和频率（或动量）的方法。该项目的应用包括波传播和其他相关现象，以及通过沿曲线的积分（x射线变换）确定函数的逆问题，以及通过边界测量确定材料结构的相关问题。虽然这个项目本身涉及他们的数学理论，但所研究的问题与物理世界密切相关。波的传播在自然界中无处不在，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，通过（最近发现的）引力波。量子粒子（如质子和电子）的散射理论是另一个由微局部分析支配的学科，微局部分析的各个方面可以用于描述远距离的量子波和半经典现象（其中普朗克常数可以被认为很小，通常在化学中）。所研究的逆问题也具有广泛的意义。这里正在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地球内部有关。这个项目的一部分描述了波在弯曲时空上的长时间或远场行为。物理上，这些出现在散射理论和广义相对论中，包括弯曲背景上的电磁波。微局部分析方法在渐近（实）双曲空间和Kerr-de - Sitter空间上的线性和非线性问题的研究中取得了突破。这里的项目旨在提高对洛伦兹散射空间的理解，其中包括渐近闵可夫斯基空间（渐近平坦空间，我们的宇宙至少在局部近似，即使有一个小的正宇宙常数）。其他项目关注边缘波的行为——特别是弹性瑞利（表面）波的衍射。另一个主要领域是逆问题，建立在最近开发的用于测地线x射线变换的空间局部反演的工具上，包括固定共形类边界刚性的解决方案：在适当的假设下，人们可以从表面上点之间的波的传播时间确定物体内部的可变声速。该领域的部分项目旨在将上述结果扩展到张量，张量在适当的意义上描述各向异性声速；也就是说，该项目将研究边界刚性，例如，从其边界距离函数中恢复黎曼度量。

This project develops and applies tools from the field of mathematics known as microlocal analysis. Roughly speaking, microlocal analysis devises methods to keep track of the position and frequency (or momentum) of waves (or, more generally, functions) simultaneously. The project's applications are to wave propagation and other related phenomena, as well as to inverse problems for determining a function from integrals along curves (the X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the project itself concerns their mathematical theory, the problems under investigation 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 is another important physical example via (the recently detected) gravitational waves. Scattering theory for quantum particles (such as protons and electrons) is another subject governed by microlocal analysis, aspects of which enter into the description of both quantum waves at large distances and semiclassical phenomena (those in which Planck's constant can be regarded as small, often the case in chemistry). The inverse problems under study are also of broad significance. One application of the theory under development 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.Parts of this project describe the long-time or far-field behavior of waves on curved space-times. Physically these arise in scattering theory and general relativity, including electromagnetic waves on a curved background. The microlocal approach to analysis on these spaces has made breakthroughs possible in work on linear and nonlinear problems on asymptotically (real) hyperbolic spaces as well as on Kerr-de Sitter space. The projects here aim to improve the understanding of Lorentzian scattering spaces, which include asymptotically Minkowski spaces (asymptotically flat spaces, which our universe approximates, at least locally, even if there is a small positive cosmological constant). Other projects concern the behavior of waves at edges -- specifically, the diffraction of the Rayleigh (surface) waves of elasticity. Yet another main area is inverse problems, building on recently-developed tools for spatially localized inversion of the geodesic X-ray transform, including the solution of fixed conformal class boundary rigidity: under suitable assumptions, one can determine a variable sound speed inside an object from the travel times of waves between points on the surface. The parts of the project in this area aim to extend the foregoing result to tensors, which describe anisotropic sound speeds in an appropriate sense; namely, the project will investigate boundary rigidity, for example, recovering a Riemannian metric from its boundary distance function.