The structure of the group of algebraic automorphisms on the affine plane $C^2$ is classically well-known. More precisely, all automorphisms on $C^2$ are tame.
Our knowledge of the structure of this group on the higher-dimensional affine spaces $C^n$ $(n ge 3)$ is very limited at present. Meanwhile we will concentrate on the 3-dimensional case. To indicate the complexity of the group $Aut (C^3)$,
it is usual to mention the Nagata automorphism $sigma$, constructed around 1972 by Nagata.
For long time it remained unknown whether or not $sigma$ is tame, untill Shestakov and Umirbaev at last proved in 2003/2004 that $sigma$ is not tame by purely algebraic method.
In this talk, we shall consider a new proof of their results from a point of view of Sarkisov Program. (The Sarkisov Program is a certain kind of algorithm to factorize
a given birational map between 3-dimensional Mori fiber spaces into simple birational maps, so-called elementary link. This framework was established finally by Corti in 1995.)
Roughly speaking, the essential lies in the maximal center of the first elementary link when applying Sarkisov Program to the Cremona transformation on $P^3$ induced by $sigma$.