Integral points on curves

The problem of finding the integral or rational solutions of polynomial equations is one of the oldest in mathematics. The simplest non-trivial case is that of one equation in two variables. While we know that for a sufficiently high degree equation there are only a finite number of solutions, an effective method for finding these solutions in general is not known to exist.

Quite recently, a far reaching program for finding the solutions using fundamental groups and p-adic numbers was invented by Minhyong Kim. I will explain some of the background, and also report on some work with Jennifer Balakrishnan and Steffen Muller using similar ideas to solve a new class of equations. I will give some concrete examples.