Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.
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The mathematical treatment of type IIB string theory belongs to algebraic geometry, specifically the deformation theory of complex structures originally studied by Kunihiko Kodaira and Donald C. Spencer.
He proved Schlessinger's theorem about representable functors of Artinian algebras and introduced Lichtenbaum–Schlessinger functors in deformation theory.