Title: Torsors on the complement of a smooth divisor and the affine Grassmannian
Abstract: A conjecture of Nisnevich predicts that for a smooth variety X over a field, a smooth divisor D
in X, and a totally isotropic reductive X-group scheme G, every generically trivial G-torsor on
X \ D trivializes Zariski locally on X. I will discuss this conjecture and related questions about
torsors under reductive groups over regular rings, as well as intervening independent questions
about the affine Grassmannian being the presheaf quotient of the loop group by the positive loop
subgroup.