Mathematics Colloquia and Seminars

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Description

Serre made two conjectures which, in essence, reduce "non-Abelian" Galois cohomology to Abelian (i.e., usual) Galois cohomology (i.e., they reduce one abstract notion to a slightly less abstract notion). When the field is the function field of an algebraic surface, the conjecture is very geometric: principal bundles over the surface for simply connected, semisimple Lie groups always admit algebraic sections. Using "rational simple connectedness", an algebraic analogue of simple connectedness replacing the interval by the projective line, and an approach suggested by P. Gille, the proof of Serre's Conjecture II was completed by A. J. de Jong, Xuhua He and myself for function fields (Conjecture I was proved long ago by Steinberg). The main lemma is a new, beautiful result for generalized Flag varieties discovered by He, and which suggests a result going beyond Serre's conjecture: the reduction map from non-Abelian to Abelian Galois cohomology is compatible with parabolic induction.