Algebraic structure of the Cremona groups and other automorphism groups

Groups of birational automorphisms of algebraic varieties is one of the oldest and quite difficult to study object of algebraic geometry. Much attention has been given both by the classics and especially recently to the study of the Cremona groups, i.e., the groups of birational automorphisms of projective spaces of dimension n. They are infinite-dimensional (for n> 1) which distinguishes them sharply from the algebraic groups most studied in algebraic geometry. Nevertheless, a number of constructions and properties of algebraic groups can be carried over to these groups. The main result consists in proving the existence of Borel subgroups in any Cremona group. This is one of the themes of the series of papers to which the talk is devoted. Also in these papers the concepts of the Jordan property of a group and its Jordan constant are introduced and studied. This caused a stream of papers both here and abroad. This technique has found applications outside of algebraic geometry. In particular, it was possible to prove the Jordan property of any connected real Lie group and, as a consequence, that of automorphism groups of many topological varieties.