I will first provide a bird-eye view upon the infrared structure of gravity. I will shortly describe the relationship between BMS symmetry, soft theorems and memory effects at leading and subleading orders in the large radius expansion, while emphasizing the specificities of super-Lorentz symmetries. Secondly, I will present a no-go result on the soft hair conjecture: supertranslations induced by matter creating and falling inside black holes do not affect Hawking radiation, though they do affect scattering amplitudes. I will start by proving that Unruh radiation is unaffected by supertranslations induced by a shockwave and then show that Hawking radiation is mathematically related to this system, as a consequence of the principle of equivalence. Third, I will explain how BMS symmetry is associated to flux-balance laws that provide constraints upon the motion of binary compact mergers. Finally, I will present the extension of the BMS group to asymptotically de Sitter spacetimes.

Symmetries are crucial in physics. Here I will develop the concept of symplectic symmetry for Einstein gravity, which is an extension of the concept of asymptotic symmetries. I will then show that the throat of the extremal Kerr black hole admits symplectic symmetries, which leads to an infinite number of conserved charges which form a Virasoro algebra in 4 dimensions and a novel generalized Virasoro algebra in higher dimensions. I will build explicitly the geometries that carry these charges, the so-called "descendants" of the extremal black hole, in the near-horizon region.