The Tame Approximation Problem for nonsolvable groups

Studying the arithmetic of homogeneous spaces of SL_n is a promising angle of attack to the Inverse Galois Problem and to local-global variants such as the Tame Approximation Problem. After giving a historical overview of this approach initiated by Emmy Noether, I will focus on recent developments on a remarkable closed subset of adelic points: the Brauer-Manin set. I will particularly show that the Brauer-Manin set is the closure of the set of rational points for homogeneous spaces of the form X=SL_n/G where G ranges through new families of nonsolvable groups, yielding to new positive answers to the Tame Approximation Problem. This is joint work with Danny Neftin.