In Terry Tao's blog, specifically this post, he says that nilpotent groups can be thought of in algebraic geometry and analysis as modeling infinitesimal perturbations of the identity. Can someone provide a reference where this point of view is used?
The idea of nilpotents representing infinitesimal change dates back at least to the invention/discovery of calculus by Newton and Leibniz! In deformation theory of schemes at least, the whole story begins with the study of flat families over the spectrum of $k[t]/t^2,$ which are viewed as infinitesimal perturbations of the "central" fibre parametrized by the nilpotent parameter $t.$ This is mentioned in Hartshorne II.8 in the exercises I believe, but there is an entire field of work in deformation theory dedicated to this beautiful concept. For example, Harthorne has a recent introductory book called "Deformation Theory".