Microlocal Analysis and Geometry

微局部分析和几何

• 批准号: 2247004
• 负责人: Andras Vasy
• 金额: $ 61.11万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247004&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Geometry

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.

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（或更普遍的功能，例如波浪的振幅和相位）。计划的应用是为了波传播和其他相关现象，以及与从表面测量结果确定材料结构以及与宇宙背景辐射成像有关的逆问题。尽管该项目涉及其数学理论，但这些问题与物理世界密切相关。波传播本质上是普遍存在的，光和引力波是重要的例子。量子颗粒的散射理论是另一个受微世分析控制的主题：这些方面进入了大距离的量子波的描述。研究中的反问题也具有广泛的意义：此处开发的理论的应用包括通过测量波的旅行时间来确定对象中弹性波的未知速度，以及通过宇宙微波背景数据的开发。许多项目都适合博士生的研究，PI努力为新一代数学家和科学家的教育做出贡献。该项目的部分描述了在弯曲的太空时间上描述了长期或遥远的野外行为，例如电磁或引力浪潮，例如电磁或引力浪潮。在这些空间上进行分析的微局部方法使PI（部分协作）在渐近和非线性问题上进行了突破性的工作，以及Kerr-de Sitter（KDS）空间（KDS）空间（宇宙学空间中旋转黑孔），​​以缓慢旋转KDS的稳定性，以证明其稳定的稳定性。最近，随着哈夫纳（Hafner）和欣兹（Hintz）的pi，其中一些工具将其中的一些工具扩展到消失的宇宙恒定案例（Minkowski，Kerr）。这里的目的是将这些工具扩展到进一步的空间，例如快速旋转的KD和Kerr SpaceTimes的扰动。项目的其他部分研究量子场理论中的基本对象，尤其是Feynman繁殖者。带有Tripathy和Zimet的新方向是在K3型表面上构建Ricci平面度量。另一个主要领域是逆问题，其中PI与Uhlmann一起引入了新的工具，用于在空间定位的地理X射线变换上进行空间定位的反转，并且Stefanov和Uhlmann将其扩展到边界刚性问题。 PI的前DostDoc Wang研究了一项项目，带有宇宙背景辐射对成像的灯光变换。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估评估来支持的。