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There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic structures:...

I have heard that for an ADE singularity $f$, $ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$ where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...