Joinings classification and applications [after Einsiedler and Lindenstruass]

This is the text accompanying my Bourbaki seminar on the work of Einsiedler and Lindenstrauss on joinings. The first five sections surveys their proof of the classification of joinings of higher-rank torus actions on arithmetic quotients of semisimple or perfect algebraic groups. The last section surveys how this classification can be used to tackle arithmetic problems, together with a new application for higher rank simple groups.