Topics in Geometrical Dynamics and Applications

本研究计画主要研究车辆运动学及相关问题。该项目的重点是一些精心挑选的具体问题与应用领域，包括追求问题，几何机器人，浮选理论和流体运动。研究的数学成果将对基本微分方程的研究产生影响（希尔方程）描述了许多自然现象，从行星运动到电子运动。研究者将积极吸引本科生和研究生参与这项研究计划。这个项目的统一主题是相关连续和离散系统的单值性，以及与有限-和无限维完全可积系统。特别是，调查员将研究车辆运动学与灯丝（副法线，局部感应，烟圈）方程，研究最多的完全可积偏微分方程之一的孤立子类型，及其离散化的连接。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.