Local systems in geometry and arithmetic

Galois laid the foundations of Galois theory and Galois groups, Riemann and Poincaré the ones of algebraic topology and the fundamental group. Grothendieck showed the two worlds are analog and developed the notion of étale fundamental group. Where do we find (continuous) representations of those in mathematics? Essentially the only ones we have at disposal are those of geometric nature. I’ll discuss conjectures on the density of those in the parameter space of all representations, and some results giving a small evidence for it (based on joint work with Moritz Kerz and Michael Groechenig).