@balarkasen Can you pass along a reference on geometric galois actions? what you were talking about above is stuff that's reminiscient of stuff i've seen when doing monodromy computations for elliptic integrals
@nick: fyi, i'm a student at the umn, and i actually know one of the grad students of the professor whose notes you linked
@Semiclassical Indeed. What I wrote above is essentially building up the Riemann surface though. The monodromy computation do indeed come to play, but that's for algebraic functions ;)
Essentially monodromies are galois groups and they "generalize" galois actions in some sense.
There are lots of works. I learned fundamental topological galois theory from Alekseev, Khovanskii, a lot is described in Szamuely but most of the works are of Grothendeick with the algebro-geometric context in his SGAs and EGAs which I don't understand yet (doubt if I ever would).
As far as I understand it, thereis a chain of generalizations Fundamental groups <--- Monodromies of Riemann Surfaces <--- Galoios groups
@Semiclassical For example, you can see that I've treated monodromies as (nonofficially, since log is not really algebraic) Galois groups in here
in official terms, it's called "anabelian geometry" or "grothedieck's galois theory" so those are the keywords you want to search with
