Differential Geometry in Cosmology

Schedule

Monday 28/11/2022, 08:30-10:30 room G
Wednesday 20/11/2022, 08:30-10:30 room C
Thursday 01/12/2022, 08:30-10:30 rrom G

Tentative program

Lecture 1. Smooth Universe, or background cosmology (Newtonian spacetime and Galilean group; Minkowski spacetime and Poincaré group; curved spacetime and diffeomorphism invariance; geodesic flow, Raychaudhuri equation; Einstein equations, conservation equations, fluid dynamics and electromagnetism in curved spacetime; cosmological principle, Robertson-Walker metric, Friedmann-Lemaître models)

Lecture 2. Real Universe, or inhomogeneous cosmology (global properties of spacetimes, Penrose representation, classification; relativistic perturbations, perturbed Einstein equations, gauge invariance; gravitational waves in empty and expanding spacetime, fluid dynamics in linear regime, cosmic microwave background radiation)

Lecture 3. Morphology of (not only) cosmic shapes (differential topology: fundamental forms, principal curvatures, mean and Gaussian curvatures, genus and Euler characteristic, Gauss-Bonnet theorem; integral geometry: Steiner theorem, Hadwiger theorem, Minkowski functionals; morphology of random fields and point processes: excursion-set model, Boolean and germ-grain models; examples: galaxy clusters, cosmic web, turbulence, porous media)