Soliton resolution for energy critical wave and wave map equations

It is widely believed that the generic dynamics of nonlinear dispersive equations in the whole space is described by solitary waves and linear dispersions. More precisely, over large times, solutions tend to de-couple into solitary waves plus radiation. It remains an open problem to rigorously establish such a description for most dispersive equations. For energy critical wave equations in the radial case, we have better understanding, using tools such as ``channel of energy inequalities" firstly introduced by Duyckaerts-Kenig-Merle, and monotonicity formulae. Recent progress in the non-radial case will also be discussed.