Title: Hermitian-Yang-Mills connections on collapsing elliptically fibered $K3$ surfaces,
Abstract: Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface with
a section, admitting a sequence of Ricci-flat metrics collapsing the fibers.
Let $\mathcal E$ be a generic, holomoprhic $SU(n)$ bundle over $X$ such that
the restriction of $\mathcal E$ to each fiber is semi-stable. Given a sequence
$\Xi_i$ of Hermitian-Yang-Mills connections on $\mathcal E$ corresponding to
this degeneration, we prove that, if $E$ is a given fiber away from a finite
set, the restricted sequence $\Xi_i|_{E}$ converges to a flat connection
uniquely determined by the holomorphic structure on $\mathcal E$.