Sormani and Wenger defined integral current spaces and the intrinsic flat distance between them. These spaces are based on the definition of integral current structure given by Ambrosio and Kirchheim. By definition, IF limits are rectifiable. In general, Gromov-Hausdorff and intrinsic flat limits need not agree and GH limits need not be rectifiable. One of the most recent advances about the interplay between GH and IF convergence is due to Matveev and Portegies. They proved that when a sequence of manifolds is noncollapsing and has a Ricci lower bound then the IF and GH limits essentially agree.
In this talk, I will go over the results about IF and GH limits coinciding. In particular, I will talk about sequences of manifolds with boundary and metric spaces satisfying the tetrahedral property and the generalised tetrahedral property.