Susanna Zimmermann

I will talk about the Cremona group of the real plane, that is, about the group of birational transformations of the complex projective plane that are defined over the real numbers. 

Specifically, I will present its Abelianisation, whose structure illustrates a totally different behavior from the Cremona group of the complex plane (defined over the complex numbers). For instance, it is not generated by a countable number of transformations and its linear maps, and every normal subgroup generated by a countable number of elements is a proper normal subgroup. The structure of the Abelianisation even implies that the those statements also hold for the subgroups of birational diffeomorphisms of the real plane, the real affine space and the sphere.