Explore the intricate world of spacetime G-structures in this third lecture by Prof. José Figueroa-O'Farrill from the University of Edinburgh's School of Mathematics. Delve into the evolution of spacetime models, from Minkowski's Lorentzian manifold to Newtonian mechanics' Galilei spacetime and Lévy-Leblond's Carroll spacetime. Examine these three spacetimes as Klein models for carrollian, galilean, and lorentzian geometries, and understand their significance in various areas of physics, including black holes, flat-space holography, and fluid dynamics. Focus on the underlying geometries, introducing Klein models, associated G-structures, and their intrinsic torsion, leading to a coarse but useful classification. If time permits, explore generalizations of these geometries, revealing Carroll and Galilei G-structures as members of a family associated with extended objects in string theory. Part of the SCREAM collaboration project, this seminar offers a deep dive into differential geometric structures and their interactions with mechanical systems, integrable systems theory, and Penrose's Conformal Cyclic Cosmology programme. Read more