W-algebras are ubiquitous extended symmetries of a vast class of CFTs, with deep connections to quantum groups and integrability. Recently, through the techniques of celestial holography, the W_{1+inf} algebra was realized explicitly in celestial correlation functions associated to 4d gravitational scattering amplitudes, but the connection to other realizations of the symmetry, and in particular integrability, was vaguely understood. In this talk I will attempt to shed some light on this issue, arguing that the emergence of the symmetry is directly connected to the color-kinematics duality relating gravity and gauge theory amplitudes, at the same time unveiling an associative structure in collinear singularities of the tree-level S-Matrix.

I will cover some of the most recent developments on classical spinning black holes and their perturbations. The reinterpretation of them in terms of a classical limit of QFT three-point amplitudes, where the black hole is modeled as a massive spinning particle, sheds light on many fundamental properties such as integrability, the Newman-Janis construction, and the so-called classical double copy relating the solution to gauge theory.