In this paper we review the regularity theory for area minimizing m-dimensional currents in codimension higher than 1, which bounds the dimension of the singular set with m − 2. In recent joint works with Emanuele Spadaro we have revisited the pioneering program of Almgren, bringing some new techniques from metric analysis and some new ideas to deal with the most intricate aspects of the proof. 157 3. First considerations in the regularity theory 165 4. The regularity theory in codimension 1

more »

... 5. Federer's theorem and the failure of ε-regularity in codimension n ≥ 2 179 6. Almgren's stratification 182 7. Multiple valued functions minimizing the Dirichlet energy 184 8. The frequency function 193 9. Approximation with multiple valued graphs 198 10. A first attempt to prove Theorem 3.4 205 11. The center manifold 209 12. The approximation on the normal bundle of the center manifold 215 13. The frequency function again 220 14. The persistence of singularities 224 References 226 2010 Mathematics Subject Classification. 49Q15, 49N60, 49Q05.