Topics in Differential Dynamics and Differential Topology

该项目涉及两个主题：台球动力系统的几何和动力学，以及接触三维流形中Legendrian结和曲线的拓扑和几何。前者包括对经典Birkhoff台球、对偶（或外）台球和投射台球的研究。 这些技术包括辛拓扑，KAM理论，Aubry-Mather理论，积分几何和符号动力学。后者包括应用“量子”拓扑勒让德和横向结和链接，研究最近推出的接触同源环及其应用，并应用斯特姆理论和理论的生成函数的全球几何的勒让德曲线。 研究台球的动机是双重的。 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中有实际应用。 其次，台球在动力系统理论中提供了一个非常好的模型，动力系统各个领域的许多发展都是由数学台球理论中的问题所激发的。 勒让德曲线理论属于两个非常活跃的研究领域的交集：辛拓扑和纽结理论。两者都与理论物理有着深刻的联系：前者与经典力学，后者与量子物理。勒让德的纽结理论为辛拓扑和纽结理论提供了一个很好的试验场，而勒让德纽结理论的进展将促进这些基础理论的新发展。

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.