  We study the general problem of equidistribution of expanding translates of
an analytic curve by an algebraic diagonal flow on the homogeneous space
$G/\Gamma$ of a semisimple algebraic group $G$. We define two families of
algebraic subvarieties of the associated partial flag variety $G/P$, which give
the obstructions to non-divergence and equidistribution. We apply this to prove
that for Lebesgue almost every point on an analytic curve in the space of
$m\times n$ real matrices whose image is not contained in any subvariety coming
from these two families, the Dirichlet's theorem on simultaneous Diophantine
approximation cannot be improved. The proof combines geometric invariant
theory, Ratner's theorem on measure rigidity for unipotent flows, and
linearization technique.
