Title: Jordan type properties for birational and biregular automorphism groups
of varieties

Abstract: Let G be a group. G is called Jordan if all of its finite subgroups
contain an Abelian subgroup of bounded index (where the bound only depends on
G). G is called bounded if all of its finite subgroups have bounded order.
Investigating Jordan type properties for birational and biregular automorphism
groups of varieties was initiated by J.-P. Serre and V. L. Popov. In this talk
we look through the history of these kind of problems. Then we discuss two
results, that the birational automorphism groups of d dimensional varieties
over fields of characteristic zero are nilpotently Jordan of class of most d,
and that most forms of flag varieties over fields of characteristic zero have
bounded biregular automorphism groups.