Victor Abrashkin (Steklov Institute and the University of Cambridge)
V.Abrashkin@dpmms.cam.ac.uk Grothendieck anabelian philosophy introduces a principle according to which the study of sufficiently general schemes over fields of finite type can be completely reduced to the study of their fundamental groups. In particular, the structure of any finitely generated over Q field can be completely recovered from the structure of its topological Galois group. In the case of local (=complete discrete valuation) fields the situation is completely different: the Galois group almost forgets the structure of the field. Nevertheless, in the case of 1-dimensional local fields one can prove an analogue of the Grothendieck Conjecture by taking into account its additional structure given by ramification filtration. We shall also discuss a generalization of this statement to the case of higherdimensional local fields.