quantum mechanics and topos theory
I can't say I understand very much of it, but The Physical Interpretation of Daseinisation
seems to present a formulation of quantum mechanics in which there's a true state space for quantum systems, which provides actual states for the system to actually be in, independent of any measurements we might want to perform on them. It then identifies classical perspectives on this state space, which represent various ways they might be measured. It turns out that this new formulation gives us an ordinary constructive logic instead of the rather intractible quantum logics theoreticians have been struggling with so far. The nice thing about constructive logics is that it's actually possible to do deductions in them, which seems not to be possible in the so-called quantum logics.
For those up to some heavy mathematics, it's neat stuff.