Abstract: I will present joint work with Jacek Jendrej (CRNS, Sorbonne Paris Nord) on equivariant wave maps with values in the two-sphere. The wave maps equation is a generalization of the free scalar wave equation to maps taking values in a manifold. The two-dimensional sphere-valued case admits stationary solutions given by finite energy harmonic maps, which are amongst the simplest examples of topological solitons (localized solitary waves with a discrete topological invariant). We prove that every finite energy equivariant wave map resolves, as time passes, into a superposition of decoupled harmonic maps and radiation, settling the soliton resolution conjecture for this equation. As a byproduct of our analysis we also prove that there are no elastic collisions between pure multi-solitons. We will discuss some of the motivations behind soliton resolution and some of the techniques involved in the proof.