Rational Points on Curves
Nils Bruin, PIMS/SFU/UBC
We will discuss the problem of finding the set of rational solutions to a polynomial equation in two variables. Such equations describe a plane curve. It has been long known that the topology of the complex solution set of such an equation largely determines the possibilities for the rational solution set. In 1985, Faltings proved Mordell's conjecture (1922) that many polynomial equations, corresponding to curves of so-called general type, admit only finitely many rational solutions. His proof does not provide a way of actually determining those solutions in any particular case, though.