Maximal connected subgroups of the Cremona group

This talk is about a work in progress with Jérémy Blanc and Andrea Fanelli. The so-called Cremona group is the group of birational transformations of the $n$-dimensional complex projective space. This group is not an algebraic group for $n > 1$, but we can hope (at least in small dimension) classify its maximal connected algebraic subgroups. In dimension $2$, the classification is old and quite easy (F. Enriques, 1893). In dimension $3$, the first rigorous treatment was done by H. Umemura in the 1980's in a series of six (quite long and technical) papers. In this talk I will explain how we can hope to recover his results in a much simpler way using the now well-developed Mori theory and discuss several possible generalizations.