The Homomorphism Form of Birational Anabelian Geometry

This project aims to explore new topics and directions in the research area of birational anabelian geometry. The main research topic in birational anabelian geometry is the study of isomorphisms
between Galois groups of finitely generated fields and the extent to which one can reconstruct isomorphisms between the fields themselves starting from an isomorphism between Galois
groups. Recently, with my collaborator Akio Tamagawa in Kyoto, we proved a new refined version of such a theory in the case of global fields: number fields, or function fields of curves over finite
fields. More precisely, we proved that the existence of an isomorphism between the three step solvable Galois groups of two global fields implies the existence of an isomorphism between the two global fields in question. This research topic aims to explore the following question, which arises naturally after the above result:
Starting from an open homomorphism between the three step solvable Galois groups of two global fields is it possible to reconstruct an embedding between the fields in question?