There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.

Most applications are physical, like electromagnetism. I wonder whether there are applications of these geometric systems beyond physics. Can you show me some active areas of research in that direction?

A related question is whether there are partial differential equations whose origin is not directly physical and that can be meaning-fully stated in terms of tensor and vector fields.

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    $\begingroup$ Differential geometry isn't outside of physics? $\endgroup$ – Qiaochu Yuan Jul 16 '12 at 18:06
  • $\begingroup$ I guess he means outside of pure maths? $\endgroup$ – Paul Reynolds Jul 16 '12 at 18:08
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    $\begingroup$ Maybe you should clarify what you mean by "beyond physics"? $\endgroup$ – Kevin Jul 16 '12 at 18:10
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    $\begingroup$ This should be Community Wiki anyhow. Voting to close until it is. $\endgroup$ – Igor Rivin Jul 16 '12 at 18:17
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    $\begingroup$ Do you count medical imaging, computer vision, pattern-recognition, computer graphics, signal-processing and control-theory as outside physics ? Possibly using discrete-differential-geometry and partial-difference equations instead of continuous versions. $\endgroup$ – user19172 Jul 17 '12 at 18:33

While I do disagree with the premises, some explicit, many implicit, of the question, the question and these premises are surely fairly popular.

While not arguing about whether differential equations and vector fields and tensors "are" physics or not, I would agree that they have huge historical/experiential base of "physical intuition", whether this is "physics" or not, notably.

In fact, it has been found profitable to transport from, or abstract from, physically meaningful situations in "mechanics", say, to "number theory" (as manifest in "automorphic forms", especially). That is, physically unsurprising, if non-trivial, ideas sometimes seem to have non-trivial potential impact on "number theory" suitably translated into harmonic analysis, understandably on special objects, not generic.

A widely-understood cliche, and wonderful it is, is the proof that $\sum 1/n^2=\pi^2/6$ via Plancherel applied to the sawtooth function made periodic, that is, on the circle as $\mathbb R/\mathbb Z$. This is easy to explain, and does touch my aesthetic sense, though I understand it might not touch others'.

If one doubted that the "guts" of differential geometry mattered, I'd volunteer that the proofs that automorphic forms do conform to expectations (about global Sobolev indices and such) do depend on the particulars, so are indeed dependent upon "geometry", whatever our conceits make of the latter.

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