Topics in Differential Dynamics and Differential Topology

微分动力学和微分拓扑主题

• 批准号: 9802849
• 负责人: Serge Tabachnikov
• 金额: $ 6.5万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802849&HistoricalAwards=false
• 关键词: Topics; Differential; Dynamics; Topology

The project concerns two topics: the geometry and dynamics of billiard-like dynamical systems, and the topology and geometry of Legendrian knots and curves in contact 3-dimensional manifolds. The former includes the study of the classical Birkhoff billiards, dual (or outer) billiards, and projective billiards. The techniques include symplectic topology, KAM theory, Aubry-Mather theory, integral geometry, and symbolic dynamics. The latter includes applications of "quantum" topology to Legendrian and transverse knots and links, study of the recently introduced contact homology rings and their applications, and applications of Sturm theory and the theory of generating functions to the global geometry of Legendrian curves. The motivation for the study of billiards is two-fold. First of all, mathematical billiards are intimately related to geometrical optics, and progress in the study of billiards may have practical applications in optics. Secondly, billiards provide a very good model in the theory of dynamical systems, and many developments in various areas of dynamical systems have been stimulated by problems from the theory of mathematical billiards. The theory of Legendrian curves belongs to the intersection of two very active research areas: symplectic topology and knot theory. Both have deep connections with theoretical physics: the former, with classical mechanics; the latter, with quantum physics. The Legendrian knot theory provides a good testing ground for symplectic topology and knot theory, and progress in the former will stimulate new developments in these fundamental theories.

该项目涉及两个主题：类台球动力系统的几何和动力学，以及接触三维流形中Legendrian结点和曲线的拓扑和几何。前者包括对经典伯克霍夫台球、对偶（或外）台球和投影台球的研究。这些技术包括辛拓扑、KAM理论、奥布里-马瑟理论、积分几何和符号动力学。后者包括“量子”拓扑在Legendrian和横向结和链路中的应用，最近引入的接触同调环及其应用的研究，以及Sturm理论和生成函数理论在Legendrian曲线整体几何中的应用。研究台球的动机有两个方面。首先，数学台球与几何光学密切相关，台球研究的进展可能在光学上有实际应用。其次，台球为动力系统理论提供了一个很好的模型，数学台球理论中的问题刺激了动力系统各个领域的许多发展。Legendrian曲线理论属于两个非常活跃的研究领域的交叉：辛拓扑和结理论。两者都与理论物理有着深刻的联系：前者与经典力学有关；后者与量子物理学有关。Legendrian结理论为辛拓扑和结理论提供了一个很好的试验场，而辛拓扑和结理论的进展将刺激这些基础理论的新发展。