The $n$-dimensional Cremona is the group of birational automorphisms of the $n$-dimensional complex projective space. This is not an algebraic group, already in dimension two. In this last case, Enriques classified all maximal connected algebraic subgroups up to conjugation.
For the three-dimensional Cremona, the same result is due to Umemura, who obtained in the 80's the full classification via algebraic group techniques.
The idea of this project is to rephrase Umemura's results via modern birational geometry and to generalise them in order to clarify the picture in higher dimension.
This is a joint work in progress with Jérémy Blanc (Basel) and Ronan Terpereau (MPIM-Bonn).