A fundamental question in the analysis of variational problems is whether functions that almost minimise a given energy, or almost solve its Euler-Lagrange equations, provide good approximations of true minimisers or critical points. In practice, one would like to understand this question not only at a qualitative level, but to obtain precise quantitative control on the distance of almost solutions to exact solutions in terms of the size of the error.

A major challenge in such a quantitative analysis of geometric variational problems is that the classical approach breaks down in the presence of singularities or changes of topology.

In this talk, we will discuss these issues for the Dirichlet energy of maps from surfaces into Riemannian manifolds for which the concentration of energy on small domains leads to a loss of compactness and the formation of a so-called bubble tree.