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Ordinary physics students are introduced to differential geometry by the lead of general relativity, but the true mathematical beauty of physics is already manifest in classical mechanics, and in a degree that is higher than general relativity, since here we encounter symplectic geometry, the theory of topological manifolds, Lie theory… Contact topology and geometry, jet bundles, groupoids and algebroids, and finally, via its natural integration with quantum mechanics, to stacks, gerbes,...