Title

Taming Infinity

Abstract

To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand

And eternity in an hour.  (William Blake)

A ramble through various mathematical topics, illustrating how group theory sometimes provides finite answers to a priori ‘infinite’ questions.

The key idea is that complicated structures can be understood with the help of symmetries. The groups of symmetries then take on a life of their own: curiosity drives us to find out as much as we can about these interesting objects.

Topics include Galois Theory, where a finite group controls the solutions of a polynomial equation, and number theory, where sometimes the infinitely many solutions of an equation over the integers can be organised into finitely many orbits of a finitely generated group.

Along the way I will mention Lie groups, which arise in geometry; arithmetic groups, which are the ‘integer points’ of Lie groups; and finite simple groups, which turn out mostly to be the analogous structures over finite fields.

Also discussed will be profinite groups, a very different kind of topological group: these are a way of ‘compactifying’ some infinite families of finite groups.