Mutation in geometry

In this thesis we link three flavors of geometry together: piece-wise linear, birational and non-commutative. The main tool to do it is the notion of mutation. In the first part we show that mutation helps us explain the structure of the Thompson group T. In the second part we study the birational geometry of mutation and get linear presentation of subgroups of the Cremona group. In the third part we plunge into derived categories and associate to every algebraic variety a triangulated category which depends only on its birational class. For rational surface this approach gives us an action of the Cremona group on the non-commutative ring. Finally we give applications of this results to the formulas of non-commutative cluster mutations.