Automorphism groups of algebraic varieties

Exploration of the automorphism (biregular or birational) groups of algebraic varieties is the classical topic of algebraic geometry. Albeit, in general, these groups are infinite dimensional (as, for example, the Cremona group $Cr (n), n>1$) some properties bring together them and the usual finite dimensional algebraic groups. The last few years are marked by increasing activity in this field accompanied by the emergence of new viewpoints. For example, in the study of finite subgroups of the automorphism (biregular or birational) groups of algebraic varieties a “social” aspect, concerning the qualitative properties of all such subgroups together, has appeared. This inspired similar studies of finite subgroups of the diffeomorphisms groups of topological manifolds. In this talk, I will discuss the results obtained in this area in recent years.