Topics in Geometrical Dynamics and Applications

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.