Orateur
Description
In 1935 Whitehead published a purported proof on Poincaré conjecture unveiling an homotopical obstruction to unique characterization of R^n among open contractible n-manifolds. However, it is now a fact that for all n>2, R^n is the unique open contractible n-manifold that is simply connected at infinity. The analogous question in algebraic geometry is to characterize the affine n-space among smooth A1-contractible affine schemes. With much novelty, this is proven in affirmation in dimensions n < 3 over fields. This question is largely open for surfaces in positive characteristics and it utterly breaks down in higher dimensions.
In this talk, we will upgrade this characterization to a Dedekind base by establishing a connection between the motivic homotopy theory and the classical affine algebraic geometry. To this end, we will see that over Dedekind scheme S, the affine n-space forms a Zariski torsor under an A^n-bundle for n < 3 and a vector bundle if S is affine. With further hypothesis, we can retrieve a similar characterization as that of fields.
