Manan was wondering about the Neukirch-Uchida-Iwasawa-Ikeda theorem: why or how does the absolute Galois group encode information about the generators of a number field?
I didn’t recognise the theorem, so wikipedia came to help: Neukirch–Uchida theorem - Wikipedia says the isomorphisms of two algebraic number fields correspond to isomorphism of its absolute Galois group. So if you know the Galois group, you can detect the specifc number field uniquely among the class of number fields.
I actually had heard of this before: when I was doing my MS thesis is found this article that extends the usual table of artihmetic topology [2604.09469v1] A Neukirch-Uchida Theorem for 3-Manifolds. Here, the Mostow-Prasad rigidity theorem for finite volume hyperbolic 3-manifolds corresponds to the NUII theorem for algebraic number fields.
Perhaps, we need a quantitative version of the NUII theorem, similar to how people study quantitative Mostow rigidity: Lathisms Lecture: Introduction to hyperbolic 3-manifolds and ideas of quantitative Mostow Rigidity - YouTube
When I pointed this out, Manan revealed he was actually lookin at an article earlier which tries to synthesise both these theorems as instances of a more general phenomena: [2508.01588] Symmetries of spaces and numbers — anabelian geometry.
The question is to reconstruct an object, among a given class perhaps, given a certain invariant of the class of objects: a group in this case of which we may consider to be “symmetries” of the object. Knots groups come to mind as an example of an invariant which is a group, which isn’t exactly symmetries of the object.
Sanskar had more things to say…