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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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Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

This is a great question! (Or rather, I quite liked it. Reasonable people can differ on this, I suppose.) In fact, local considerations (or even semi-local considerations like understanding the symple …
Dustin Connery-Grigg's user avatar