# Abstract Roman Fedorov

**Roman Fedorov ** will speak on

** Around the Grothendieck-Serre conjecture on principal bundles**

Abstract: Let X be a connected regular scheme, G a reductive group scheme over X, and E a principal G-bundle over X . A conjecture of Grothendieck and Serre predicts that E is trivial locally in Zariski topology, if it is trivial over some Zariski open subset of X. The conjecture was proved by Ivan Panin and the speaker, when X is a scheme over an infinite field; the case of schemes over finite fields was settled by Panin later.

We will discuss different formulations of the conjecture, its relation to homotopy invariance of the first non-abelian cohomology functor, and outline the strategy of the proof.