Research
Here are some slightly more specific (and more technical) blurbs describing things I am currently looking at:
Anosov representations have come to be considered a good generalization of convex cocompactness in the higherrank setting. Is there a corresponding notion of geometric finiteness?

Teichmüller space has many useful metrics on it. What about higher Teichmüller spaces?
In the case of the Hitchin component BridgemanCanaryLabourieSambarino defined a pressure metric, using dynamical invariants associated to the geodesic flow on the group, and there may be a complexanalytic interpretation of this metric, by workinprogress of DeroinTholozan.
Here is a different idea: can we define an analogue, for higher Teichmüller spaces, of Thurston's (asymmetric) Lipschitz metric on Teichmüller space?

Is CAT(0) the "natural" / "best" notion of NPC in the context of groups?
Under this fuzzy umbrella comes tricky questions like "Are all Gromovhyperbolic groups CAT(0)?", the BradyBestvina wager (True or false: the fundamental group of a NPC manifold is Gromovhyperbolic iff it contains no higherrank free abelian subgroup), and so on.

Where is the boundary between CAT(0) and cubulated?