Examples of applications in diophantine approximations

Structure of quotient spaces of Lie groups, lattices in semisimple Lie groups, Mahler compactness criterion, ergodiicty and mixing via unitary representations, results of Mautner, Howe and Moore, nondivergence for unipotent flows, statement of Ratner's theorems and proofs of special cases, Dani-Smillie and Dani-Margulis linearization technique, statements of measure rigidity results of Einsiedler-Katok-Lindenstrauss and Benoist-Quint