Abstract: ?
Homogeneous dynamics is a special kind of dynamical systems ?given by Lie groups. For certain important cases, the systems are ?ergodic and hence the points with nondense orbits form a set of ?measure zero. However, the behavior of nondense orbits reflects the ?complexity of the system and is related to certain important ?number-theoretic problems. In this talk, we will discuss some recent ?progresses of investigations in nondense orbits of homogeneous ?dynamics, with emphasis on properties of bounded orbits. We will also ?explain their relations with problems in number theory.

About speaker:
Professor Jinpeng An now works ?at the Institute of Mathematical Sciences of Peking University. His ?research focuses on the lie theory and dynamics of group Actions. ?Great progress has been made in the function of subgroups in ?homogeneous space and the approximation problem of diophantine graphs. ?In particular, it has been proved that the two-dimensional weighted ?inferior approximation vector set is the resultant set. Therefore, a ?new proof of Schmidt's conjecture is given and stronger results are ?obtained. In cooperation with other scholars, they proved some ?properties of subgroup dimension data, solved two lie group problems ?proposed by famous mathematician Langlands, applied the results to ?spectral geometry, and proved single connected compact Riemannian ?manifolds with equal spectrum but different embryos.

摘要: ?
齐次动力系统是由李群给出的一类特殊动力系统。对特定的重要情形，系统是遍历的且具有非稠密轨道的点构成一零测集。然而，非稠密轨道的行为反映了系统的复杂性并和一些重要数论问题有关。本次报告，主讲人将讨论齐次动力系统非稠密轨道的最新研究进展，重点讲述有界轨道的性质。同时，主讲人还将在报告中解释该系统与数论问题的关系。

报告人简介: ?
安金鹏教授现就职于北京大学数学科学学院，主要研究领域为李群及群作用的动力学。安教授在齐性空间上的子群函数和丢番图图的逼近问题上取得了重要进展。特别是证明了二维加权内逼近向量集为致胜集，从而给出了Schmidt猜想的新证明，并得到了更强的结果。与人合作证明了子群维数数据的若干性质，解决了著名数学家Langlands提出的两个李群问题，并将结果应用到谱几何，证明了存在等谱但不同胚胎的单连通紧无边黎曼流形。