The seminal work of Brezis-Coron for 2-dimensional harmonic maps introduced an estimate that leads to the existence of minimisers of the Dirichlet energy in different homotopy classes. This had important consequences in the study of 3-dimensional harmonic maps, leading to the celebrated result of Tristan Rivière about the existence of everywhere discontinuous harmonic maps and the partial regularity result of Hardt-Lin-Poon for minimisers of the axially symmetric relaxed Dirichlet energy.
We will discuss what analogies and differences arise when following a similar path for the 1-dimensional half-harmonic map case and for the 4-dimensional Yang-Mills functional.
The talk will be based on joint works with Ali Hyder (TIFR Bangalore) and Tristan Rivière (ETH Zurich), and it is supported by Fondazione Cariplo and CDP. | 
