We study the following real version of the famous Abhyankar-Moh Theorem: Which real rational map from the affine line to the affine plane, whose real part is a non-singular real closed embedding of $\mathbb{R}$ into $\mathbb{R}^2$, is equivalent, up to a birational diffeomorphism of the plane, to the linear one? We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are plenty of non-equivalent smooth rational closed embeddings up to birational diffeomorphisms. Some of these are simply detected by the non-negativity of the real Kodaira dimension of the complement of their images. But we also introduce finer invariants derived from topological properties of suitable fake real planes associated to certain classes of such embeddings.
(Joint Work with Adrien Dubouloz).[-]

The Brauer group of a del Pezzo or a K3 surface over a number field is thought to govern the existence of rational points. A large piece of this group is determined by the Galois-module structure on the geometric Picard group of a surface. I will present work in progress that, given equations for a low-degree del Pezzo or K3 surface, determines its algebraic Brauer group with a high degree of confidence. I will also indicate how e˙ective versions of the Chebotarev density can certify probabilistic results, under GRH. Technology permitting, I will show a live demo.N.B. This is joint work with Austen James.[-]