Study of hyperbolic maps and Anosov flows is an old and well-developed area of research in Dynamical System. Partial hyperbolicity is a natural generalization of hyperbolicity. It is a relatively new field which gained traction since 1990's due to its strong connection with ergodic theory and most importantly, its own essence. Extensive research has been done in this area in the last three decades, but there is a lot more left to explore.
At this moment I am thinking on quasigeodesic Anosov flows in three dimension. A flow is called 'quasigeodesic' when its flowlines are length minimizing up to some multipicative and/or additive bounds.
I am also interested about geometric group theory and related topics, specially dynamical and rigidity properties of various classes of groups.
New Classes of Quasigeodesic Anosov Flows; with Sergio Fenley. (ArXiv)
Leafwise Quasigeodesic Foliation in Dimension Three and the Funnel Property with Sergio Fenley. Ergodic Theory and Dynamical Systems (Arxiv)
Topological Shadows and Complexity of Islands in Multiboundary Wormholes ; with a group of physicists.