Space-Time Structure Reconstruction in Cosmology and Lorentzian Geometry

这项研究旨在开发用于宇宙学和爱因斯坦广义相对论的数学工具。更具体地说，该项目打算推进从观测到的天体物理数据中恢复引力波、黑洞和宇宙弦等时空结构的技术。一种类型的数据是宇宙微波背景(CMB)，通过NASA的威尔金森微波各向异性探测器和欧洲航天局的普朗克探测器项目进行了高精度的测量。CMB各向异性包含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。这一结果有望也适用于运动器官的医学成像分析。这项研究将为研究生培训和跨学科合作提供机会。这项研究包含几个项目，涉及从积分几何到偏微分方程式、微局部分析和洛伦兹几何的一系列数学科目。第一个项目处理洛伦兹几何中的积分几何或层析成像问题，该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。研究人员将重点研究变换的内射性、稳定性和微局域性质等重要问题，以解决洛伦兹几何中反问题的应用，如散射刚性问题。第二个项目将通过探索光线变换和双曲型微分方程组理论之间的联系来确定由Sachs-Wolfe效应产生的原始引力波。第三个项目将侧重于使用基于线性Boltzmann方程的动力学模型的CMB反问题。此外，该项目旨在从光线变换的微观局部稳定性来探索运输型方程的稳定性，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 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.