abstract: We will cover the theory of unipotent flows, and its applications to number theory. We will start with quantitative recurrence results for unipotent flows. These results, on their own, lead to the solutions of many old problems in the matric theory of diophantine approximations. We will cover some of the applications (such as a proof of the Mahler conjecture), and subsequent developments.
We will then proceed to the measure classification results on unipotent flows (Ratner's theorem). We will give a proof of Ratner's theorem in a simple case, and then cover some of the applications. In particular we will give a proof of the Oppenheim conjecture (we may also present Margulis' original proof which does not rely on Ratner's theorem). We will then cover results on the quantitative Oppenheim conjecture, and some (both older and more recent) applications to diophantine equations. We will end by presenting very recent results of Margulis and Goetze on quadratic forms in five or more variables.