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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

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Theoretical physics: Why not just $\mathbb{R}^4$?

You and I are having a conversation: "Okay," I say, "I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles." "Something like that." "...And the …
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