At the end of the nineteen century Norwegian mathematician
Sophus Lie (1842-1899), together with Swedish mathematician Albert
Bäcklund (1845-1922), established a theorem that in modern terms reads
as follows: any Lie transformation (resp., field) of some domain of the
jet space J^{k}(n,m) is either the lift of a contact transformation
(resp., field), if m=1, or the lift of a point transformation (resp.,
field), if m>1. The proof of this pivotal result in the geometric theory
of PDEs, which I will carry out in this seminar, gives us the
opportunity to examine in depth some features of the jet spaces and
their natural structures.