Microlocal Analysis and Applications

计划的研究开发和应用微局部分析领域的工具。粗略地说，这个场同时跟踪波的位置和频率，或动量（或更一般地说，函数）。计划应用于波传播和其他相关现象，以及从沿曲线的积分（x射线变换）确定函数的反问题，以及从边界测量确定材料结构的相关问题。虽然这个建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（不久前发现的）引力波：PI最近与Hintz的合作表明，在一个具有（可能很小，正如我们目前所理解的）正宇宙常数的宇宙中，黑洞的扰动衰变为一个略有不同的黑洞，在这个过程中发射引力波。量子粒子（如质子和电子）的散射理论是由微局部分析支配的另一个主题：这些方面都进入到远距离量子波的描述中。所研究的反问题也具有广泛的意义：这里发展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弹性波的未知变速，例如，这与利用地震波的传播时间成像到地球内部有关。许多项目适合博士生进行研究，PI努力为新一代数学家和科学家的教育做出贡献。有些计划描述波在弯曲时空上的长时间或远场行为。从物理上讲，这些都出现在广义相对论中，包括弯曲背景上的电磁波。分析这些空间的微局部方法使得PI（与他以前的博士生P. Hintz）在渐近双曲（AH）空间以及克尔-德西特（KdS）空间（宇宙时空中的旋转黑洞）的线性和非线性问题上的工作取得了突破，最终与Hintz一起证明了缓慢旋转KdS空间的稳定性。最近，与Hafner和Hintz一起，PI将这些工具中的一些扩展到宇宙常数消失的情况（Minkowski, Kerr）。这里的项目旨在将这些工具扩展到更远的空间，比如克尔时空的摄动，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一起引入了用于测地线x射线变换的空间局部反演的新工具，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各向异性弹性，这在地球内部起着重要作用。另一个与PI前博士后合作的项目是研究宇宙背景辐射成像中光线变换的潜在应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.