Seeing Through Space-Time - IPAM at UCLA

Explore the intricacies of inverse problems in spacetime and cosmic microwave background radiation in this comprehensive lecture by Gunther Uhlmann at IPAM's Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy Tutorial. Delve into the determination of spacetime structure through light observation sets and gravitational waves, examining topology, differential structure, and conformal metrics. Investigate the inverse problem of determining early gravitational perturbations from CMB measurements, focusing on the light ray transform. Gain insights into nonlinear hyperbolic equations, Lorentzian manifolds, and the mathematical foundations of gravitational wave astronomy. Covers topics such as Einstein rings, conical refraction, causal relations, geometrical optics, and microlocal analysis, providing a deep dive into the mathematical challenges of modern cosmology and astrophysics.

Introduction
Motivation
Nonlinearity
Earthquakes
Active Measurements
Ultrasound
Passive Measurements
Gravitational Lenses
Einstein Ring
Conical Refraction
gravitational waves
model
Lorenzian geometry
Causal Relations
Observations
Conformal Structure
Active Measurement
Geometrical Optics
Nonlinear ways
Quadratic nonlinearity
The proof
Quadratic case
Microlocal analysis
Subsonic singularity