Space-Time Structure Reconstruction in Cosmology and Lorentzian Geometry

Emory University, Atlanta GA

Investigators

Abstract

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.