Interior regularity of area-minimizing currents with high codimension

Integral currents are a weak generalization of smooth oriented manifolds with boundary and provide a natural setting in which to study the Plateau problem: ‘what are the surfaces of least m-dimensional area that span a given (m-1)-dimensional boundary?’ However, the weak nature of integral currents permit the formation of singularities. The problem of determining the size and structure of the interior singular set of an area-minimizer in this setting has been studied by many since the 1960s, with many ground breaking contributions. The codimension one case is significantly easier to handle, but in the higher codimension case, much less progress has been made since the celebrated (m-2)-Hausdorff dimension bound on the singular set due to Almgren, the proof of which has since been simplified by De Lellis and Spadaro.

In this talk I will review the key features of the proof of Almgren/De Lellis-Spadaro and discuss how to strengthen this to an upper Minkowski dimension estimate. I will also discuss some work in progress with Camillo De Lellis (IAS) towards establishing rectifiability of the singular set in high codimension.