The polihomogeneity of the gravitational field and conserved charges at null infinity

In this talk we'll give an overview of the conformal Einstein field equations (CEFEs) and briefly discuss some of the global non-linear stability results obtained with the CEFEs in asymptotically flat and de-Sitter-like spacetimes. Motivated the caveats that emerge from these results, we'll discuss the problem at spatial infinity and its impact (polyhomogeneity) on the peeling behaviour of the gravitational field. To give a simple but illustrative discussion of the so-called cylinder at spatial infinity (i^0), we'll focus the analysis on the polyhomogeneity of spin-0 fields in Minkowski spacetime propagating close to the cylinder at i^0 and the calculation of conserved charges at the critical sets where spatial and null infinity meet.