Title: Equidistribution of expanding translates of curves in homogeneous spaces and Diophantine approximation. Abstract: We consider an analytic curve $\varphi: I \rightarrow \mathbb{M}(n\times m, \mathbb{R}) \hookrightarrow \mathrm{SL}(n+m, \mathbb{R})$ and embed it into some homogeneous space $G/\Gamma$, and translate it via some diagonal flow $A=\{a(t): t > 0 \} < \mathrm{SL}(n+m,\mathbb{R})$. Under some geometric conditions on $\varphi$, we prove the equidistribution of the evolution of the translated curves $a(t)\varphi(I) in G/\Gamma$ as $t \rightarrow \infty$. As an application, we prove that for almost all points on the curve, the Dirichlet's theorem can not be improved. This is a joint work with Nimish Shah.