Frans Oort

Abstract:   We discuss properties of the topological and of the algebraic fundamental group of an algebraic curve. The talk will be a survey of topics discussed in a current seminar at MIT. Grothendieck showed that Galois theory and the theory of the topological fundamental group can be viewed inside one concept: the fundamental group. We discuss this definition, and we give applications: a criterion for good reduction for algebraic curves (Takayuki Oda), isogenies of hyperbolic curves (Mochizuki), the Grothendieck anabelian conjecture (Nakamura, Tamagawa, Mochizuki). All topics in the talk are contained in published papers; we will give a survey of these ideas, and try to make them understandable for a general audience.