Let the set O={non-degenerate, indefinite, real quadratic forms in 3-variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{|Q(v)|: v is a non-zero integral vector in $R^3$}. The set M={m(Q): Q is in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis' proof of the Oppenheim conjecture asserts that, for every a>0 $M /intersect (a, /infty)$ is a finite set. In this lecture we will show that #{M /intersect (a, /infty)}<<a^{-26}.

This is a joint work with Prof. Margulis, and our method is based on dynamics on homogeneous spaces.

令集合$O$为所有"关于3个变量的非退化, 非定的实系数二次型"组成的集合. 对于$O$中二次型$Q$, 我们定义其Markov下界$m(Q)=inf{|Q(v)|^3/|det(Q)|: v∈Z^n-{0}}$, 这里$det(Q)$为$Q$对应的对称矩阵的行列式. 我们称非负实数集${m(Q): Q∈O}$为3维Markov谱. 由Cassels-Swinnerton-Dyer早年的工作和Margulis对Oppenheim猜想的证明可知, 对于每个正实数$a>0$, $M∩(a, ∞)$为一个有限集合. 在报告中我们将介绍Markov谱的计数问题的最新进展:$#M∩(a, ∞)<<a^{-26}$.

齐性空间动力系统技术为问题的主要研究方法, 报告的结果是基于和Margulis教授的合作.