Regularity of area minimizing currents mod $p$

We establish a first general partial regularity theorem for area minimizing currents $\mathrm{m}\mathrm{o}\mathrm{d}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{m}\mathrm{o}\mathrm{d}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure.