Microlocal Analysis and Applications

计划中的研究开发和应用微观局部分析领域的工具。粗略地说，这个场同时跟踪波的位置和频率或动量（或更一般地说，函数）。计划的应用是波的传播和其他有关现象，以及从沿沿着曲线的积分（X射线变换）确定函数的逆问题和从边界测量确定材料结构的有关问题。虽然这个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（不久前发现的）引力波：PI最近与Hintz的工作表明，在一个（可能很小，因为我们的宇宙目前被理解）正宇宙学常数的宇宙中，黑洞的扰动衰减为一个稍微不同的黑洞，在这个过程中发射引力波。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：这些方面都进入了对大距离量子波的描述。研究中的逆问题也具有广泛的意义：这里开发的理论的应用是通过测量波的传播时间来确定物体中弹性波的未知可变速度，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。许多项目适合博士生研究，PI致力于为新一代数学家和科学家的教育做出贡献。一些拟议的项目描述了弯曲时空上波的长时间或远场行为。从物理上讲，这些现象出现在广义相对论中，包括弯曲背景上的电磁波。微局域方法在这些空间的分析中取得了突破，使PI在渐近双曲（AH）空间和克尔-德西特（KdS）空间（宇宙时空中的旋转黑洞）的线性和非线性（与他以前的博士生P. Hintz）问题上的工作成为可能，最终证明了Hintz缓慢旋转的KdS空间的稳定性。最近，PI与Hafner和Hintz一起将其中一些工具扩展到消失的宇宙学常数情况（Minkowski，Kerr）。这里的项目旨在将这些工具扩展到更远的空间，例如克尔时空的扰动，以及研究宇宙时空的其他方程。其他项目研究量子场论中的基本对象，特别是费曼传播子。另一个主要领域是逆问题，PI与Uhlmann一起引入了空间局部化反演测地线X射线变换的新工具，并与Stefanov和Uhlmann一起将其扩展到边界刚性问题。这里的一个项目旨在将其扩展到在地球内部起重要作用的各向异性弹性。另一个项目与PI的前博士后王研究的光线变换与潜在的应用程序的成像的宇宙背景辐射。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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 is another important physical example via (the not long ago detected) gravitational waves: the PI’s recent work with Hintz showed that in a universe with a (possibly small, as our universe is currently understood) positive cosmological constant, perturbations of black holes decay to a slightly different black hole, emitting gravitational waves in the process. 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. The inverse problems under study are also of broad significance: an application of the theory developed here is the determination of an unknown variable speed of elastic waves 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. Many of the projects are suitable for research by doctoral students, and the PI strives to contribute to the education of a new generation of mathematicians and scientists.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. The microlocal approach to analysis on these spaces has made breakthroughs possible in the PI's work on linear and non-linear (with his former PhD student, P. Hintz) problems on asymptotically hyperbolic (AH) spaces as well as Kerr-de Sitter (KdS) space (rotating black holes in a cosmological spacetime), culminating in the proof of the stability of slowly rotating KdS spaces with Hintz. More recently, with Hafner and Hintz the PI extended some of these tools to the vanishing cosmological constant case (Minkowski, Kerr). The projects here aim to extend these tools to further spaces, such as perturbations of Kerr spacetimes, and also to study other equations on cosmological spacetimes. Other projects study basic objects in quantum field theory, in particular the Feynman propagator. Another main area is inverse problems, where the PI, together with Uhlmann, has introduced new tools for the spatially localized inversion of the geodesic X-ray transform, and with Stefanov and Uhlmann extended this to the boundary rigidity problem. One project here aims to extend this to anisotropic elasticity which plays an important role in the interior of the Earth. Another project with the PI's former postdoc Wang studies the light ray transform with potential applications to imaging by the cosmic background radiation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.