Conformal geometry and related topics 1100-CGRT

Conformal geometry is concerned with the study of spaces or spacetimes equipped with a structure that allows one to measure angles, but not arc lengths. Many differential equations in mathematical relativity, notably those connected to massless particles, are conformally invariant. In fact, a number of powerful theorems in mathematical relativity, such as the Robinson theorem, the Goldberg-Sachs theorem and the Kerr theorem, find a more natural formulation in conformal geometry.

From a purely mathematical perspective, a number of important geometric structures on Riemannian and pseudo-Riemannian manifolds, such as Hermitian and Robinson manifolds, are conformally invariant. Conformal geometry also arises from other types of related geometries, such as Cauchy--Riemann (CR) geometry, projective geometry and generic vector distributions.

This course is intended to Master-level and PhD-level students. The topics covered include the basics of conformal geometry, its associated calculus, known as tractor calculus, and the Fefferman-Graham ambient metric. In addition, we shall discuss its relation to twistor geometry and underlying geometric structures, such as totally null complex vector distributions and CR geometry.