Research

Symmetries are the one of the most useful tools in mathematics, physics and engineering. Motion of a rigid body, elastic beams or even the visual cortex V1 can be described using dynamical systems on Lie groups, which make rigorous our intuitive notion of a symmetry. That is why those that are special in some sense systems are of a great interest and of a great importance in pure and applied math.

One particular source of such models can be obtained from classical mechanics. Take, for example, a spherical robot with a camera in its shell. Can you flip the camera view by rolling the sphere around? Or more precisely is there a control that takes the sphere to the same point but with a different orientation? The answer is yes, but one would like to have a good algorithm that would assign to each configuration the corresponding control. There are many ways to do it, of course, but there is one particularly nice example that I have found together with prof. Sachkov via optimal control techniques. Other examples on which I worked personally is the sub-Riemannian problem on SO(3) (with Yu. Sachkov) and Engel structures on Lie groups (with A. Medvedev).