It is not too hard to show that if two integers $m$ and $n$ can both be written as a sum of two squares, so can $mn$. This is a consequence of a more general result, there is a natural group structure on the set of $SL(2,\mathbb{Z})$ equivalence classes of integer binary quadratic forms of a fixed discriminant. In this talk we will look at some ways of understanding this structure and applications. Speaker(s): Andy Gordon (UM)