in 1966 shafarevich introduced the notion of “infinite dimensional algebraic group”, shortly “ind-group”. his main application was the automorphism group of affine $n-space \ a^n$ for which he claimed some interesting properties. recently, jointly with j.-ph. furter we showed that the automorphism group of any finitely generated (general) algebra has a natural structure of an ind-group, and we further developed the theory. it turned out that some properties well-know for algebraic groups carry over to ind-groups, but others do not. e.g. every ind-group has a lie algebra, but the relation between the group and its lie algebra still remains unclear. as another by-product of this theory we get new interpretations and a better understanding of some classical results, together with short and transparent proofs. an interesting “test case” is $\ aut(\ a^2)$, the automorphism group of affine 2-space, because this group is the amalgamated product of two closed subgroups which implies a number of remarkable properties. e.g. a conjugacy class of an element $g \in \ aut(\ a^2)$ is closed if and only if $g$ is semi-simple, a result well-known for algebraic groups. a generalization of this to higher dimensions would have very strong and deep consequences, e.g. for the linearization problem. note: there will be a $pre-talk$ for graduate 2022年亚洲世界杯预选赛 from 2:30-3:00. the speaker has kindly accepted to tell our graduate 2022年亚洲世界杯预选赛 what an ind-group is. the regular talk will begin at 3:00.