p-adic numbers and applications

Let p be a prime number. Then the p-adic numbers were discovered by Hensel in the early part of the twentieth century, as the Cauchy completion of the rational numbers Q with the respect to a certain metric, the so-called p-adic metric. Shortly afterwards, Ostrowski proved that any ``reasonable" metric on the rational numbers is equivalent to either a p-adic metric, for some prime p, or the usual Archimedean metric on the reals. Thus the p-adic numbers are in some sense no more or no less natural than the usual Archimedean metric.

A pervading and fruitful theme in number theory is to study arithmetic questions from a p-adic viewpoint, and in this talk we will discuss the origin of the p-adics and some recent and not-so-recent results that come from p-adic methods.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).