Microlocal Analysis of Linear and Nonlinear Problems

该项目开发和应用被称为微观局部分析的数学领域的工具。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。该项目的应用是波的传播和其他相关的现象，以及逆问题确定一个功能的积分沿着曲线（X射线变换）和相关的问题，确定结构的材料从边界测量。虽然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是普遍存在的，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学例子，通过（最近探测到的）引力波。量子粒子（如质子和电子）的散射理论是另一个受微局域分析支配的学科，它的各个方面都进入了对大距离量子波和半经典现象（普朗克常数可以被认为是小的，通常是化学中的情况）的描述。所研究的反问题也具有广泛的意义。该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，这与使用地震波的传播时间对地球内部进行成像有关。该项目的部分内容描述了弯曲时空中波的长时间或远场行为。在物理上，这些现象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。微局部的方法来分析这些空间取得了突破，可能在工作中的线性和非线性问题的渐近（真实的）双曲空间，以及对克尔德西特空间。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（渐近平坦的空间，我们的宇宙至少在局部近似，即使有一个小的正宇宙学常数）。其他项目涉及波在边缘的行为-特别是弹性瑞利波的衍射。另一个主要领域是逆问题，建立在最近开发的工具上，用于测地X射线变换的空间局部化反演，包括固定共形类边界刚度的解决方案：在适当的假设下，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。该项目在这一领域的部分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;即，该项目将研究边界刚度，例如，从其边界距离函数恢复黎曼度量。

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.