Surfaces and p-adic fields

The philosophy of arithmetic topology, first established by Mazur, gives an analogy relating arithmetic to lower dimensional topology. Under this philosophy, one gets a dictionary, relating between Number fields and 3-manifolds, primes and knots, and p-adic fields and surfaces. In the talk I will try and explain why these surprising analogies were drawn. I will also outline some recent work of the speaker, which further establishes this connection for the local case, using tools such as graphs of groups and Bass–Serre theory.