Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) algebraic geometry.

Besides its applications in physics theory, I want to know if there are math motivations to study supergeometry. I've seen some papers and monographs on supergeometry, but most of them are trying to follow classical even geometry: they are just trying to establish super analogues of theorems in even geometry.

I want to ask:

- Are there any results in even geometry that are no longer true in the super case?
- Are there any new phenomena that only happen in the super (purely odd or mixed) case?
- Are there any results in geometry that were first discovered in the super setting?

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