Microlocal Analysis and Applications

微局部分析及应用

基本信息

批准号：
1953987
负责人：
Andras Vasy
金额：
$ 37.77万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953987&HistoricalAwards=false
关键词：
Microlocal Analysis Applications

项目摘要

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.

计划的研究发展并应用了微局部分析领域的工具。粗略地说，该领域简单地跟踪波（或更一般的功能）的位置和频率或动量。计划的应用是为了浪潮传播和其他相关现象，以及沿曲线积分（X射线变换）的函数的逆问题，以及与边界测量结果确定材料结构的相关问题。尽管该提议涉及其数学理论，但这些问题与物理世界紧密相关。波传播本质上是普遍存在的，电磁波（例如光）是最普遍的例子之一。一般相对性理论是另一个重要的物理示例（不久前检测到的）引力波：PI最近与Hintz的工作表明，在一个具有（可能很小的宇宙，我们当前理解的宇宙）的宇宙中，宇宙学的正常存在，黑洞的扰动，黑色孔衰减到一个略有不同的黑洞，发射了剧烈的重力波。量子颗粒的散射理论（例如质子和电子学）是由微局部分析控制的另一个受试者：这些方面都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是通过测量波的旅行时间来确定物体中未知的弹性波的变化速度，例如使用地震波的旅行时间与地球内部的成像相关。许多项目适合博士生的研究，PI努力为新一代数学家和科学家的教育做出贡献。一些拟议的项目描述了在弯曲的太空时间上波浪的长期或远处的野外行为。从物理上讲，这些是在一般相对论中出现的，包括弯曲背景上的电磁波。 PI在线性和非线性（他的前博士学位学生，P。Hintz）的工作中，对这些空间进行分析的微局部分析方法成为了突破性的突破性问题。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）。这里的项目旨在将这些工具扩展到进一步的空间，例如KERR空间的扰动，并研究其他关于宇宙学空间的等式。其他项目研究量子场理论中的基本对象，尤其是Feynman繁殖者。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。这里的一个项目旨在将其扩展到各向异性弹性，该弹性在地球内部起着重要作用。 PI的前DostDoc Wang研究了另一个项目，灯光变换具有宇宙背景辐射对成像的潜在应用。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响评估标准来通过评估来通过评估来获得珍贵的支持。