Supersymmetry in Mathematics

Needless to say, Justice Stewart did not have in mind supersymmetry; but I believe that his words apply here as well. Of course, it is easy enough to define it, but doing so may actually obscure its message. My aim instead is to help you know it when you see it.

I would like to argue that one of the mathematical signatures of supersymmetry is that it relates first-order to second-order conditions, and that often the first-order conditions imply a certain limiting case of the second-order conditions.

I will now discuss several examples taken from Physics and Geometry. I would be very interested in learning about new instances of this phenomenon. Please get in touch if you think you have identified one, for I believe that some sort of supersymmetry will lurk behind it.

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"I can't define it, but I know it when I see it."
— Justice Stewart in Jacobellis v. Ohio (paraphrase)