Monodromy representations and arithmetic

Daniel Litt (IAS)
Fine Hall 322

Let X be a smooth complex variety. Which representations of the fundamental group of X come from geometry? Motivated by this question, I'll sketch the proof of an analogue of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields, as well as a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero.