Microlocal Analysis and Geometry

该项目开发和应用微观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（或更一般地说，函数，如波的振幅和相位）。计划中的应用是波的传播和其他相关现象，以及与从表面测量确定材料结构以及宇宙背景辐射成像有关的逆问题。虽然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，光波和引力波就是重要的例子。量子粒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包括通过测量波的旅行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确定宇宙的发展。许多项目都适合博士生进行研究，PI致力于为新一代数学家和科学家的教育做出贡献。项目的部分内容描述了弯曲时空上的波（如电磁波或引力波）的长时间或远场行为，包括波的存在。微局域方法在这些空间的分析上取得了突破，使PI（部分合作）在渐近双曲空间以及Kerr-de Sitter（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Kerr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。Tripathy和Zimet的一个新方向是在K3型曲面上构造Ricci平坦度量。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。该奖项反映了NSF的法定使命，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。

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.