Topics in Geometrical Dynamics and Applications

本课题研究车辆运动学及其相关问题。该项目专注于一些精心挑选的具体问题，其应用领域包括追求问题、几何机器人、浮选理论和流体运动。这项研究的数学结果将对描述从行星运动到电子运动等许多自然现象的基本微分方程（希尔方程）的研究产生影响。研究者将积极地让本科生和研究生参与这个研究项目。这个项目的统一主题是相关的连续和离散系统的单一性，以及与有限维和无限维完全可积系统的强烈联系。特别地，研究者将研究车辆运动学与灯丝（双正规，局部感应，烟圈）方程的联系，灯丝方程是研究最多的完全可积的孤子型偏微分方程之一，以及它的离散化。该研究将有助于离散微分几何和离散完全可积系统的新兴领域。

This research project studies vehicle kinematics and related problems. The project focuses on a number of carefully selected concrete problems with applications in areas including pursuit problems, geometrical robotics, flotation theory, and fluid motion. The mathematical results of the research will have an impact on the study of fundamental differential equations (Hill's equation) that describe numerous natural phenomena, from planetary motion to the motion of electron. The investigator will actively involve undergraduate and graduate students in this research program.The unifying theme of this project is the monodromy of the related continuous and discrete systems and a strong connection with finite- and infinite-dimensional completely integrable systems. In particular, the investigator will study connections of vehicle kinematics with the filament (binormal, local induction, smoke ring) equation, one of the most studied completely integrable partial differential equations of soliton type, and its discretizations. The research will contribute to the emerging areas of discrete differential geometry and discrete completely integrable systems.