This talk surveys the classification of joinings of higher-rank torus actions on S-arithmetic quotients of semisimple or perfect algebraic groups and some of its applications. This classification was proved by Einsiedler and Lindenstrauss (Duke Mathematical Journal 2007, Publications mathématiques de l’IHÉS, 2019). It establishes that ergodic joinings must be algebraic, and in particular that such torus actions in many cases must be disjoint, that is, they admit only the trivial joining which is the product of the Haar measures on each of the factors.
Their proof is based on entropy methods, developed by Einsiedler, Katok, Lindenstrauss and Spatzier. We will describe these methods and give some ideas on how they fit into the scheme of their proof. Specifically, we will explain how to prove disjointness when the associated algebraic groups have a different root structure. This already allows for some applications, which will be presented at the end of the talk.
