The so-called Sarkisov program aims to decompose any birational map between two Mori fibre spaces into a sequence of elementary transformations, called Sarkisov links.
This tool is extremely useful in studying groups of birational self-maps of algebraic varieties. In this talk, I will first discuss some recent results and open questions in the Sarkisov program for algebraic surfaces over non-closed fields (including Severi-Brauer surfaces). Then I will show how these results imply some spectacular properties of higher Cremona groups. Based on joint works with J. Blanc and J. Schneider. | 
