One critical difficulty in the study of geometric nonlinear wave equations, compared with the classical NLW, is that there are nontrivial interactions between unbalanced frequencies. This requires one to understand a class of linear paradifferential flows, which describe the evolution of high frequency waves on a low frequency background. This talk continues and completes the stuff in the August workshop lectures. A brief review of the workshop notes might be helpful in order to better place this talk in the broader context.