Roughly, I am interested in categorical dualities in logic. Here are some more specific questions I sometimes think about.

• In our recent paper Dirk and I proposed the idea of dialgebraic logic, which provide a coalgebra-like framework for modal logics. All examples we gave are modal extensions of intuitionistic logics. I'm wondering if there are more logical paradigms that fit the framework of dialgebraic logic.
• Yosida duality is a duality for compact Hausdorff spaces with certain Riesz spaces. I'm curious if there is a natural description of the Yosida dual of the Vietoris functor and whether this gives rise to an interesting modal logic. (The Vietoris functor is of course well known from its appearance in modal logic over a classical base.)
• More to come