A geodesic is the generalization of a straight line in Euclidean space to Riemannian manifolds. From a physical point of view, a geodesic on a Riemannian manifold can be thought of as the trajectory of a particle in the absence of any forces. Similarly, one can determine the trajectory of a charged particle on a Riemannian manifold in the presence of a magnetic field. Now the particle is subject to a single force, the Lorentz force, and the resulting trajectory is called a magnetic geodesic. In my disseration, I studied the length spectrum and topological entropy of the magnetic geodesics on nilmanifolds with a left-invariant Riemannian metric and left-invariant magnetic field.

I also think about questions related to "best" metrics on Lie groups. Among all left-invariant Reimannian metrics, one can ask which ones have the most symmetry. I am interested in understanding which Lie groups admit metrics of maximal symmetry and which do not. In the case of the former, what are they?