Dispersion and collapse of wave maps

We study numerically the Cauchy problem for equivariant wave maps from 3 + 1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behaviour in the formation of black holes. AMS classification scheme numbers: 35L67, 35L70, 35Q75.