Abstract: We consider the canonical spectral triple associated to a two-torus as a compact Riemannian spin manifold. As an example for the local index formula in the even case, we compute the index of the corresponding Dirac operator with coefficients in a rank one vector bundle given by a special class of projections.

Abstract: In the early 90's, Bogomolov introduced a program whose ultimate goal is to reconstruct function fields of dimension $> 1$ over algebraically closed fields from their pro-$\ell$ 2-step nilpotent Galois groups. Although it is far from being resolved in full generality, this program has since been carried through for function fields over the algebraic closure of a prime field. After an introduction to birational anabelian geometry and Bogomolov's program, in this talk I will describe the possibility of a $Z/\ell$-analogue, including the inherent problems/difficulties, as well as some partial results in this direction.