Mappings of finite distortion: analysis in the extreme

Degenerate structures arise naturally in many questions in PDE's and calculus of variations as well as in their applications. Mappings of finite distortion study how far the powerful tools of geometric analysis can here reach; the theory of these maps has been developed in the last 7-8 years, and it is surprising how precise information we can now achieve. We give an illustration on these methods through two examples, one in nonlinear elasticity and the other in inverse problems, in electrical impedance tomography.