Geometric and Topological Study of Systems with Impact and Related Topics

提出的研究涉及具有冲击的动力系统的几何和拓扑：经典台球、磁场台球、外台球和其他由变分原理或几何结构定义的系统。主要研究周期轨道的数量和结构、系统的稳定性、可积动力学的刚性以及相关的几何和拓扑问题。这些技术包括变分原理（特别是莫尔斯理论）和代数拓扑和几何的相关方法，特别是各种构型空间的拓扑研究。应用于其他领域，特别是机器人运动规划问题的拓扑问题。所提议的研究的主要重点是在看似遥远的思想和方法之间的相互关系，从几何学和拓扑学，一方面，和动力系统，力学，电动力学和机器人，另一方面。该提议的研究将通过宾夕法尼亚州立大学数学高级研究学期计划（MASS）对数学本科研究产生影响，PI是该计划的主任。

The proposed research concerns the geometry and topology of dynamical systems with impact: classical billiards, billiards in magnetic fields, outer billiards and other systems defined by variational principles or geometrical constructions. The research focuses on the number and structure of periodic orbits, stability properties of these systems, rigidity of integrable dynamics and related problems of geometry and topology. The techniques include variational principles (in particular, Morse theory) and related methods of algebraic topology and geometry, in particular, topological study of various configuration spaces. Applications to other areas are concerned, in particular, topological issues of the motion planning problems in robotics.The main emphasis of the proposed research is in the interrelations between seemingly distant ideas and approaches from geometry and topology, on the one hand, and dynamical systems, mechanics, electrodynamics and robotics, on the other. The proposed research will make an impact on undergraduate research in mathematics via the Mathematics Advance Studies Semesters program (MASS) at Penn State of which the PI is the Director.