Diophantine properties of affine subspaces and their submanifolds

\begin{abstract}
It is a classical theorem of Khintchine-Groshev that $\R^n$ obeys a
Diophantine $0-1$ law with respect to Lebesgue measure. One can ask
the same question for submanifolds of $\R^{n}$. This turns out to be
more difficult. I will speak about a result which shows that a large
class of affine hyperplanes, as well as their non-degenerate
submanifolds exhibit Groshev type behavior. The proof is a variation
of a method of Kleinbock-Margulis and involves dynamics on
$\SL(n,\R)/\SL(n,\Z)$. If time permits, I will outline possible
generalizations.
\end{abstract}