The Cremona group of rank n over a field k may be defined algebraically as the group of k-automorphisms of the field of rational functions in n variables with coefficients in k, or geometrically as the group of birational automorphisms of the projective space of dimension n over k. In rank 1, one just gets the automorphism group of the projective line, PGL(2,k). But the structure of the Cremona group in higher rank is very involved. Its study began in 1863 and was first developed via methods of projective algebraic geometry. It has seen a new impetus after 2000, with ramifications along many directions based on different techniques:

• Birational geometry
• Geometric group theory
• Algebraic transformation groups
• Automorphisms of algebraic varieties
• Motivic techniques
• Dynamical viewpoint.

Among the recent results, let us single out

• The proof by Cantat and Lamy of the non-simplicity of the Cremona group of rank 2 viewed as an abstract group (2013),
• The proof of the corresponding result in higher rank by Blanc, Lamy and Zimmermann (2021),
• The construction of a CAT(0) cubical complex equipped with an action of the Cremona group by isometries, and many applications, by Lonjou and Urech (2021),
• The construction of a dominant rational map of the projective plane having a transcendental dynamical degree, by Bell, Diller and Jonsson (2020).

The subject presents a great variety of open problems, of various levels of difficulty. But its approach and the interaction between the experts are being made more difficult by the increasing diversity of the tools and methods. The school aims at giving a unified presentation of the subject, starting with the basic notions and results of its main ramifications.

Participation to the conference is subject to registration. Registration will be possible on this website from Monday, March 3 to Friday, May 2.

Jérémy Blanc (Université de Neuchâtel), Serge Cantat (CNRS-Université de Rennes), Anne Lonjou (Université Paris Saclay).

Michel Brion, Emmanuel Peyre.