  For any integer $n\ge 2,$ $0\le j\le n-2,$ any sequence of (not necessarily
connected) smooth orientable compact Riemannian manifolds $N_j$ of dimension
$j,$ and any closed compact subsets $K_j$ in $N_j$, we construct a smooth
compact Riemannian manifold $M^{2n-k+1}$, with a smooth calibration $\phi$ and
a calibrated irreducible homologically area minimizing surface $T^n$ in $M$.
($k$ is the smallest number with $K_k$ nonempty.) The singular set of $T$ is
the disjoint union of $K_0,\cdots,K_{n-2},$ and the $j$-symmetric part of the
$j$-th strata in the Almgren stratification of the singular set of $T$ is
precisely $K_j.$ As a corollary, area-minimizing integral currents can have
fractal singular sets. Thus, we settle a conjecture by Almgren and the answer
is sharp dimension-wise. As a by-product, we also obtain similar conclusions
for mod $p$ area-minimizing surfaces and stable minimal surfaces.
