Arithmetic Homogeneous Spaces

Ideal classes in (totally real) number fields give naturally rise to compact orbits inside SL(n,Z)\SL(n,R) for the diagonal subgroup. We will discuss their (equi-)distribution properties as the field varies, and the two main ideas in our approach: bootstraping diophantine estimates to an entropy statement (Linnik's method), and the measure rigidity for higher rank torus actions. This is joint work with Lindenstrauss, Michel, and Venkatesh.