Oxford

I will present a universal definition of the integers in the field of rational numbers, building on work discussed by Bjorn Poonen in his seminar last term. I will also give, via model theory, a geometric criterion for the non-diophantineness of Z in Q.

Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data.
In algebraic geometry, faithfully flat descent says that if $X\rightarrow Y$ is a faithfully flat morphism of schemes, then giving a sheaf on $Y$ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $X$, satisfying certain compatibility conditions. Translated to algebra, it says that if $S\rightarrow R$ is a faithfully flat morphism of rings, then giving an $S$-module is the same as giving a certain simplical module over a simplicial ring constructed from $R$. In topology, given an etale cover $X\rightarrow Y$ one can recover $Y$ (at least up to homotopy equivalence) from a simplical space constructed from $X$.