In this talk we discuss the question of quantitative stability of minimisers for the classical Dirichlet energy of maps into the unit sphere, i.e. whether, and with what rate, the distance of a map which almost minimise the energy (with given degree) to the nearest minimiser can be bounded in terms of the energy defect.
We will see that there is a marked difference between maps of degree 1 and maps of higher degree and will discuss how a more flexible approach to quantitative stability and specially designed gradient flows can be used to establish sharp quantitative stability results for maps for which energy concentrates at multiple scales and/or near multiple points.