Abstract: This talk will be a gentle introduction to the relationship between rational points of an algebraic curve and the structure of its (algebraic) fundamental group. For hyperbolic curves over a finitely generated infinite field, Grothendieck conjectured (in 1983) a precise relationship between fundamental groups and rational points, which we will explain by analogy with the corresponding problem in topology. I will then discuss the degree to which Grothendieck's conjecture holds for the "generic curve of type (g,n)". For such curves, the structure of the mapping class group of type (g,n) plays a central role.
