Research
Here are some slightly (not very much) more specific and technical blurbs describing things I am currently looking at:-
Anosov representations have come to be considered a good generalization of convex cocompactness in the higher-rank setting. Is there a corresponding notion of geometric finiteness?
How useful will such a notion be in understanding degenerations of Anosov representations? How useful will it be in understanding representations which are like Anosov representations in many but not all ways (e.g. have limit maps with good properties, but may contain unipotents?)
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Teichmüller space has many useful metrics on it. What about higher Teichmüller spaces?
In the case of the Hitchin component Bridgeman-Canary-Labourie-Sambarino defined a pressure metric, using dynamical invariants associated to the geodesic flow on the group, and there may be a complex-analytic interpretation of this metric, by work-in-progress of Deroin-Tholozan.
Here is a different idea: can we define an analogue, for higher Teichmüller spaces, of Thurston's (asymmetric) Lipschitz metric on Teichmüller space?
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Is CAT(0) the "natural" / "best" notion of NPC in the context of groups?
Under this fuzzy umbrella comes tricky questions like "Are all Gromov-hyperbolic groups CAT(0)?", the Brady-Bestvina wager (True or false: the fundamental group of a NPC manifold is Gromov-hyperbolic iff it contains no higher-rank free abelian subgroup), and so on.
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Where is the boundary between CAT(0) and cubulated?