Anabelian geometry

Anabelian geometry is a theory in number theory, which describes the way to which algebraic fundamental group G of a certain arithmetic variety V, or some related geometric object, can help to restore V. The first traditional conjectures, originating from Alexander Grothendieck and introduced in Esquisse d'un Programme were about how topological homomorphisms between two groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki. Before anabelian geometry proper began with the famous letter to Gerd Faltings and Esquisse d'un Programme, the Neukirch–Uchida theorem hinted at the program from the perspective of Galois groups, which themselves can be shown to be étale fundamental groups. Anabelian geometry is a theory in number theory, which describes the way to which algebraic fundamental group G of a certain arithmetic variety V, or some related geometric object, can help to restore V. The first traditional conjectures, originating from Alexander Grothendieck and introduced in Esquisse d'un Programme were about how topological homomorphisms between two groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki. Before anabelian geometry proper began with the famous letter to Gerd Faltings and Esquisse d'un Programme, the Neukirch–Uchida theorem hinted at the program from the perspective of Galois groups, which themselves can be shown to be étale fundamental groups.