Abstract: It is known that if a finite energy, sphere-valued wave map develops a singularity, it must do so by concentrating the energy of (possibly) several copies of the ground state harmonic map at a point in space. If only a single bubble of energy is concentrated, the solution decomposes into a dynamically rescaled harmonic map plus a term that encodes the energy that radiates away from the singularity. In a breakthrough work, Krieger, Schlag, and Tataru proved that such singular dynamics do occur. In this talk we revisit the solutions they constructed, showing that they can be built from the part of the map that radiates away from the singularity. We give a sharp classification of the dynamical blow-up rate for every solution with this prescribed radiation. This is joint work with Jacek Jendrej and Casey Rodriguez.