  In the present work, we consider area minimizing currents in the general
setting of arbitrary codimension and arbitrary boundary multiplicity. We study
the boundary regularity of 2d area minimizing currents, beyond that, several
results are stated in the more general context of $(C_0, \alpha_0, r_0)$-almost
area minimizing currents of arbitrary dimension $m$ and arbitrary codimension
taking the boundary with arbitrary multiplicity. Furthermore, we do not
consider any type of convex barrier assumption on the boundary, in our main
regularity result which states that the regular set, which includes one-sided
and two-sided points, of any 2d area minimizing current $T$ is an open dense
set in the boundary.
