Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills problems

We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S^2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We derive sharp asymptotics on the dynamics at the blow up time and prove quantization of the energy focused at the singularity.