A proof of the geometric case of a conjecture of Grothendieck and Serre concerning principal bundles

This talk is about our joint work with Roman Fedorov. Assume that $U$ is a regular scheme, $G$ is a reductive $U$-group scheme, and $\mathcal{G}$ is a principal $G$-bundle. It is well known that such a bundle is trivial locally in étale topology but in general not in Zariski topology. A. Grothendieck and J.-P. Serre conjectured that $\mathcal{G}$ is trivial locally in Zariski topology, if it is trivial at all the generic points.
We proved this conjecture for regular local rings $R$, containing infinite fields. Our proof was inspired by the theory of affine Grassmannians. It is also based significantly on the geometric part of a paper of the second author with A. Stavrova and N. Vavilov.