There is a rich interplay of dynamics on locally homogeneous spaces and number theory. For instance, Oppenheim's conjecture on the denseness of values of quadratic forms was proved by Margulis using dynamics of a
unipotent one-parameter subgroup (that preserves the form). Ratner's rigidity theorems covers more generally actions of groups generated by unipotent flows and has found many more applications.
The dynamics of the diagonal subgroup of SL(3,R) by right multiplication on SL(3,Z) \ SL(3,R) (and similar higher rank Cartan actions) is less understood but has similar connections to number theory. For instance, Littlewood's conjecture on Diophantine approximation follows from Margulis' conjecture on invariant measures for that action. In joint work with A. Katok and E. Lindenstrauss we have obtained partial results on these conjectures.
We will discuss the above types of actions and applications.