On regularity for integral currents - the smooth approximation of cycles

This thesis is devoted to the study of regularity properties of generalized surfaces in the Plateau problem. The first part of the thesis focuses on the regularity theory for Federer-Fleming integral currents, proving that every integral cycle can be approximated by a smooth submanifold up to a singular set of codimension 5. This is based on a joint study with William Browder and Camillo De Lellis, completing the program started by the former author with Frederick J. Almgren. The second part of the thesis extends the previous result to unoriented domains, showing that integral cycles mod 2 can be approximated by smooth submanifolds up to a singular set of codimension 3; in addition, this estimate on the singular set can be refined depending on the codimension of the cycle. Each estimate on the singular set is optimal, and the submanifolds can be taken free of singularities if the homology class admits a smooth embedded representative.