It is by now well understood how leading soft theorems follow as Ward identities of asymptotic symmetries defined at null infinity. For subleading infrared effects the connection is more subtle, but it turns out that this can be formalised, to all orders in the energy expansion, by adapting the Stuckelberg procedure to construct an extended radiative phase space at null infinity. I will exemplify this with Yang-Mills theory, showing the construction of the extended phase space, as well as the charges corresponding to the subleading soft theorems at all orders. These turn out to satisfy simple recursion relations, and organise themselves into infinite dimensional algebras in certain subsectors.

I will show that self-dual gravity in Euclidean four-dimensional Anti-de Sitter space (AdS4 ) can be described by a minimally coupled scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalisation of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS4 version of the so-called kinematic algebra. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS4 by expanding around the self-dual sector. Moreover, I will show that the new kinematic Lie algebra can be lifted to a deformed version of the w1+∞ algebra, which plays a prominent role in celestial holography.

The idea of writing supergravity as a double copy of super Yang-Mills theories has proved a fruitful one, most notably in the context of scattering amplitudes. I will explore this idea of "squaring" at a fundamental level to explain how the symmetries of the former(both local and global) can be written as a double copy of the latter. I will show how the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry are obtained from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincare. We give a gravity-gauge dictionary by convoluting fields and parameters. At the global level, I will explain how the coset groups of supergravity can be derived from the global R-symmetries of the SYM multiplets, via the 4 division algebras:reals, complexes, quaternions and octonions. Finally I will explore some intriguing possible applications of our dictonary.