Space-Time Structure Reconstruction in Cosmology and Lorentzian Geometry

这项研究旨在开发宇宙学和爱因斯坦广义相对论中使用的数学工具。更具体地说，该项目旨在推进从观测到的天体物理数据中恢复时空结构（如引力波、黑洞和宇宙弦）的技术。一种类型的数据是宇宙微波背景（CMB），它是通过美国航天局的威尔金森微波各向异性探测器和欧洲航天局的普朗克勘测者项目等项目进行高精度测量的。宇宙微波背景辐射的各向异性包含了有关宇宙早期状态及其演化的重要信息。研究人员将研究恢复这些信息的数学理论，并开发稳定的成像方法和统计推断方法。预计结果也适用于运动器官的医学成像分析。该研究将为研究生培训和跨学科合作提供机会。该研究包含几个项目，涉及一系列数学科目，从积分几何到偏微分方程，微局部分析和洛伦兹几何。第一个项目涉及洛伦兹几何中的积分几何或层析成像问题，涉及从零测地线上的积分恢复函数或张量，称为光线变换。研究人员将专注于重要的问题，如注入性，稳定性和微局部性质的变换，以解决应用逆问题的洛伦兹几何，如散射刚性问题。第二个项目将是通过探索光线变换和双曲型微分方程理论的联系，从萨克斯-沃尔夫效应中确定原始引力波。第三个项目将侧重于使用基于线性玻尔兹曼方程的动力学模型的CMB逆问题。此外，该项目旨在从光线变换的微局部稳定性探索传输型方程的稳定性，并将分析扩展到非线性问题。该奖项反映了NSF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。

This research aims to develop mathematical tools of use in cosmology and Einstein’s general relativity theory. More specifically, the project intends to advance techniques to recover spacetime structures such as gravitational waves, black holes, and cosmic strings from observed astrophysical data. One type of data is the Cosmic Microwave Background (CMB), which was measured to high precision through NASA projects such as Wilkinson Microwave Anisotropy Probe and European Space Agency project Planck Surveyor. The CMB anisotropies contain important information regarding the early state of the universe and its evolution. The investigator will study the mathematical theory for recovering such information and develop stable imaging methods and statistical inference methods. The results are anticipated to also be applicable in medical imaging analysis of moving organs. The research will provide opportunities for graduate student training and interdisciplinary collaborations. The research contains several projects involving a range of mathematical subjects from integral geometry to partial differential equations, microlocal analysis, and Lorentzian geometry. The first project deals with an integral geometry or tomography question in Lorentzian geometry, which concerns recovering a function or tensor from its integral over null geodesics, called the light ray transform. The investigator will focus on important questions such as the injectivity, stability, and microlocal properties of the transform to address applications to inverse problems in Lorentzian geometry, such as the scattering rigidity problem. The second project will be the determination of primordial gravitational waves from the Sachs-Wolfe effects by exploring the connection of the light ray transform and the theory of hyperbolic type differential equations. The third project will focus on the CMB inverse problem using kinetic models based on the linear Boltzmann equation. In addition, the project aims to explore the stability of transport type equations from the microlocal stability of the light ray transform and extend the analysis to nonlinear problems.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.