Conformal Geometry with a View to Understanding the Cosmos

In general relativity, the universe is often formalized as a four-dimensional ``space-time”, i.e. a smooth manifold equipped with a signature (3,1) pseudo-Riemannian metric. I give an introduction to general relativity, motivating this formalism. We then discuss conformal compactifications of space-times, and observe that conformal geometry can be a more convenient approach to studying space-times than pseudo-Riemannian geometry. Finally we introduce conformal tractor calculus and outline its use in studying the asymptotic curvatures of certain hypersurfaces in space-times.