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?
- Morse theory for the uniform energy, I. Adelstein and J. Epstein, J. Geom. 108 (2017), no. 3, 1193-1205.
- Topological entropy of left-invariant magnetic flows on 2-step nilmanifolds, J. Epstein, Nonlinearity. 30 (2017), no. 1, 1-12.