Currents with coefficients in groups and applications

The mass-minimization problem for 1-dimensional rectifiable currents with coefficients in a group provides flexible tools for the solution of other classical problems in optimization of 1-dimensional objects. For instance, the Steiner tree problem consists in finding the shortest connected set including some fixed points; we show how this problem can be solved as a mass-minimization problem for 1-dimensional rectifiable currents with coefficients in a suitable group. Thanks to the representation adopted for these currents, we can exploit the calibration method and we can analyse some examples. Other applications are the Gilbert-Steiner problem and the problem of modelling dislocations in crystals.