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  • $\begingroup$ If you were studying PDE on $\mathbb{R}^n$ you might also care about classifying manifolds because of the homotopy principle. By the same token, the topological K-theory led to the theory for C*-algebras, which in turn (for example) provided a complete classification for AF-algebras: see, e.g. tinyurl.com/yj99wpe (wiki article, the asterisk is troublesome here) $\endgroup$ Commented Feb 24, 2010 at 19:33
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    $\begingroup$ With all respect to h-principle, my guess is that most PDE people do not care about it. Nor do I think that having a classification of say simply-connected manifolds help solve any PDE. $\endgroup$ Commented Feb 24, 2010 at 20:04