Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

In my talk I will survey some of the developments regarding the classification of measures on arithmetic quotients of semisimple groups invariant under diagonalizable subgroups (i.e. tori). I will then considering an application that is not so new but gives a good illustration of the power and limitations of these techniques obtained jointly with Einsiedler, Michel and Venkatesh, regarding the distribution of Q tori with some splitting conditions in an arithmetic quotient arising from the unit group of a division algebra.