Carrollian and celestial spaces at infinity

We show that the geometry of asymptotic infinities Minkowski spacetime (in $d+1$ dimensions) is captured by homogeneous spaces Poincar\'e group: blow-ups spatial (Spi) and timelike (Ti) in sense Ashtekar--Hansen a novel space Ni fibering over $\mathscr{I}$. embed these \`a la Penrose--Rindler into pseudo-euclidean signature $(d+1,2)$ as orbits same subgroup O$(d+1,2)$. describe corresponding Klein pairs determine their Poincar\'e-invariant structures: carrollian structure on Ti, pseudo-carrollian Spi "doubly-carrollian" Ni. give additional geometric characterisations grassmannians affine hyperplanes spacetime: (double cover the) grassmannian lorentzian hyperplanes; Ti spacelike fibers null planes, which exhibit fibred product $\mathscr{I}$ lightcone celestial sphere. also total bundle scales conformal symmetry algebra its doubly-carrollian isomorphic to $\mathscr{I}$; is, BMS algebra. how reconstruct from any geometries, establishing points parametrise certain cuts geometries. include an appendix comparing with (A)dS observe de Sitter groups have no could play r\^ole sphere plays flat holography.