In the mid 1980's, Margulis completed the proof of the Oppenheim conjecture: If $Q$ is a non-degenerate, indefinite real quadratic form in n-variables (n>2), and is not proportional to a quadratic form of rational coefficients. Then for every $a>0$ there is a non-zero integral vector v in $R^n$, such that $|Q(v)|<a$.

Margulis's proof uses dynamics on homogeneous spaces of Lie groups, which seems unrelated to the above problem. It is worth mentioning that homogeneous dynamics is essentially the only method, up to now, to prove the Oppenheim conjecture in its full generality (methods in analytical number theory can only handle the cases for n>4, but not for n=3, 4). In this lecture, we shall discuss the main idea of the proof, and also some related results in the geometry of numbers.

二次型是关于一些变量的二次齐次多项式. 二次型理论与几何, 群论, 数论等数学分支都有着紧密的联系. 对于一个实系数的二次型, 如果它不能表示成一个有理系数二次型的倍数, 我们则称其为无理二次型. 在20世纪80年代中期, Margulis证明了关于无理二次型的Oppenheim猜想: 假设Q为关于n个变量的(n>2)非退化, 非定(既非正定亦非负定)的实系数无理二次型. 则对任意小的数$a>0$, 存在$R^n$中的一个非零的整向量$v∈Z^n-{0}$满足$|Q(v)|<a$.

Margulis的证明运用了(和问题本身看似无关的)齐性空间动力系统技术. 值得一提的是, 齐性空间动力系统是目前彻底证明Oppenheim猜想的唯一途径. 经典数论的办法只能证明n>4的情形, 而不能处理n=3或4时的情况. 我们将在报告中介绍Margulis证明Oppenheim猜想的主要思路, 以及与此相关的几何数论中的结果.