Quantization of time-like energy for wave maps into spheres

In this talk, we shall discuss how building upon the threshold theorem for wave maps, techniques inspired by the blow-up analysis of supercritical harmonic maps, can lead to a decomposition of the map into a decoupled sum of rescaled solitons, along a suitably chosen sequence of time slices converging to the maximal time of existence, with a term having asymptotically vanishing energy in the interior of the light cone, and when the target manifold is an Euclidean sphere. This work is motivated by the soliton resolution conjecture, on which spectacular progress has been achieved recently for equivariant wave maps, radial Yang-Mills fields and semi-linear critical wave equations.