Geometric calculations using Grassmann variables

Physicists seem to get huge computational value by introducing Grassmann variables and Grassmann integration into differential geometric calculations.

Can someone here motivate these techniques mathematically, and include the simplest pure-math example where their use and value can be illustrated. I have thought they were invented for computing volumes of constrained moduli spaces, although physicists did not originally describe them this way. Do any mathematicians use them rigorously to actually solve problems?

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Grassman variables are a neat and handy way to talk about exterior algebras and the geometric objects built on them, like forms and spinors. The physics notation in particular gets a lot of mileage out of Grassman calculations that look formally like Gaussian integrals and behave remarkably like them. The best intro to that, and the bet purely mathematical application I know, is Matthai and Quillen's paper where they get an explicit representation of the Thom form that is quite powerful. They can be a little intimidating because physicists speak about them in language grating to mathematicians and tend to use them in the vicinity of highly nonrigorous thinking, but they themselves can be worked with mathematically without difficulty.

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