Microlocal Analysis and Geometry

该项目开发并应用了微局部分析领域的方法。粗略地说，微局部分析同时跟踪波的位置和频率或动量(或者更广泛地说，函数，例如波的幅度和相位)。计划的应用是波传播和其他相关现象，以及与通过表面测量确定材料结构以及宇宙背景辐射成像有关的反问题。尽管该项目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，光和引力波就是重要的例子。量子粒子的散射理论是微局域分析的另一个主题：这些方面进入了对大距离量子波的描述。正在研究的反问题也具有广泛的意义：这里发展的理论的应用包括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，以及通过宇宙微波背景数据来研究宇宙的发展。许多项目适合博士生研究，PI努力为新一代数学家和科学家的教育做出贡献。该项目的部分内容描述了弯曲时空中波的长期或远场行为，包括电磁波或引力波的存在。这些空间的微局域分析方法使得PI在渐近双曲空间以及Kerr-de Sitter(KDS)空间(宇宙时空中旋转的黑洞)上的线性和非线性问题的工作(部分是协作的)取得了突破，最终证明了具有Hintz的缓慢旋转的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常量情况(Minkowski，Kerr)。这里的目的是将这些工具扩展到更远的空间，例如快速旋转的KDS和克尔时空的微扰。该项目的其他部分研究量子场论中的基本对象，特别是费曼传播子。一个新的方向是在K3型曲面上构造Ricci平坦度量，这是Tripary和Zimet的一个新方向。另一个主要领域是反问题，PI与Uhlmann一起引入了新的工具来进行测地线X射线变换的空间局域反演，并与Stefan ov和Uhlmann一起将其扩展到边界刚性问题。与PI的前博士后王一起研究光线变换及其潜在应用于宇宙背景辐射成像的项目。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project develops and applies methods in the area of microlocal analysis. Roughly speaking, microlocal analysis keeps track of the position and frequency, or momentum, of waves (or more generally, functions, such as the amplitudes and phases of waves) simultaneously. The planned applications are to wave propagation and other related phenomena, as well as inverse problems related to determining the structure of a material from surface measurements as well as to imaging by cosmic background radiation. Although the project concerns their mathematical theory, these problems are closely connected to the physical world. Wave propagation is ubiquitous in nature, with light and gravitational waves being important examples. Scattering theory of quantum particles is another subject governed by microlocal analysis: these aspects enter into the description of quantum waves at large distances. The inverse problems under study are also of broad significance: applications of the theory developed here include the determination of an unknown variable speed of elastic waves in an object via the measurement of travel times of waves, as well as the development of the universe through cosmic microwave background data. 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.Parts of the project describe the long-time or far field behavior, including existence, of waves, such as electromagnetic or gravitational waves, on curved space-times. The microlocal approach to analysis on these spaces has made breakthroughs possible in the PI's (in part collaborative) work on linear and nonlinear problems on asymptotically hyperbolic 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 aim here is to extend these tools to further spaces, such as fast rotating KdS and perturbations of Kerr spacetimes. Other parts of the project study basic objects in quantum field theory, in particular the Feynman propagator. A novel direction, with Tripathy and Zimet, is construction of Ricci flat metrics on K3-type surfaces. 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. A 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.