I work in differential geometry and dynamical systems. These two fields have many connections. The geodesic flow in particular introduces dynamical techniques to study geometric problems. For example, finding closed geodesics is equivalent to finding periodic orbits of the geodesic flow. Does every compact Riemannian manifold have infinitely many closed geodesics? For the case of the two-dimensional sphere, this age old problem was resolved using ideas from low-dimensional dynamics in the late 90s.
Most of my work concerns rigidity properties of both Riemannian manifolds and dynamical systems. This area is has developed from the classical theorems due to Mostow and Margulis in the 1960s and 70s. The Strong Rigidity Theorem by Mostow is the paradigm: In dimension at least 3, the fundamental group of a closed manifold of constant negative curvature determines its isometry type. Ideas and techniques from differential geometry, group theory and dynamical systems were all essential to the proof. Margulis greatly strenghtened this result in his superrigidity and arithmeticity theorems under the much more stringent higher rank assumption that the manifolds in questions have lots of totally geodesic flat subspaces.
These results have inspired many works. In geometry, there are various characterizations of symmetric spaces in terms of simple geometric properties. For example, non-positively curved closed manifolds have to be locally symmetric if every geodesic is contained in a totally geodesic flat subspace, much in the spirit of Margulis’ theorem. The latter also was generalized to certain symmetric spaces with negative curvature, and some non-Riemannian spaces of higher rank.
In dynamics, these results inspired the investigation of actions of higher rank abelian and semi-simple groups. Surprisingly, suitable such actions are locally rigid, i.e. cannot be perturbed, they do not have time changes and few invariant probability measures. Some of these actions incorporate phenomena from number theory where the scarcity of invariant measures has given rise to important results.
If you want to get a better feeling for this area of research, I will be glad to talk with you and/or direct you to some introductory reading, e.g. my recent survey An Invitation to Rigidity Theory.