Semi-calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of semi-calibrated currents. We will focus mostly on the case of dimension $2$, where it turns out that semi-calibrated cycles are actually pseudo holomorphic. By using an analysis implementation of the algebro-geometric blowing up of a point we study the regularity of semi-calibrated $2$-cycles from the point of view of uniqueness of tangent cones and of local smoothness.

Constrained versions of the mean curvature flow have shown their unfriendliness due to the global nature of the equation, which makes the usual techniques in extrinsic flows either fail (like comparison principle, preservation of embeddedness,...) or become more subtle. Despite the difficulties, such flows (specially those preserving area or enclosed volume) are still quite appealing because they are specially well suited for applications to the isoperimetric problem. This is an overview talk of the current situation of the study of such flows; we will review what has been done and which are the perspectives for the future. This will finish with the presentation of two counterexamples for the preservation of mean convexity (which has been claimed to be preserved in a couple of papers) and positivity of the scalar curvature, resp. This ends the hope of doing a singularity analysis á la Huisken-Sinestrari for such constrained flows.