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Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers in a certain area of mathematics thought that their work could be of value to some field other than mathematics—maybe it was some kind of engineering, so I'll just call it "engineering"—but then it was found that interactions between engineers and mathematicians made substantial contributions to mathematical research but not to engineering. I don't remember what it was about, beyond that.

So my question is: What are the most edifying examples in recent centuries, of applications to fields other than mathematics greatly benefitting mathematical research when mathematicians had expected to be only the benefactors of those other fields?

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    $\begingroup$ Examples of which phenomenon? (The whole first paragraph sets the stage so vaguely that it is not helpful.) $\endgroup$
    – Matt F.
    May 4 at 18:34
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    $\begingroup$ Related: mathoverflow.net/questions/14782/… $\endgroup$ May 4 at 21:42
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    $\begingroup$ The simplex method in linear programming motivated a lot of the systematic study of convex polytopes. $\endgroup$ May 4 at 21:48
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    $\begingroup$ Among others.. Von Karman and Jhukovsky, Fourier, Mohr, KF Gauss contributed to maths propelled from fields of aerodynamics, electrical engineering and structural mechanics/stress analysis, land surveying/geodesy respectively. $\endgroup$
    – Narasimham
    May 4 at 21:57
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    $\begingroup$ @SamHopkins I might be misinterpreting Michael Hardy's question, but it sounds to me that he's asking for more than examples of "pure math spinoffs from the solution to an applied math problem." The challenge there would be to find examples of math that did not spin off from an applied problem if you go back far enough historically. Rather, I think he wants examples of "math meets engineering, no significant engineering progress is made, but math is greatly enriched by the encounter." Linear programming wouldn't count, then, because it is "too successful" engineering-wise. $\endgroup$ May 5 at 15:24

4 Answers 4

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If I may substitute "physics" for "engineering", one could argue that string theory is an example of a topic where mathematicians have interacted with a different field of research and the dominant benefit of that interaction was in mathematics (witness the award of a Fields medal to a physicist).

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  • $\begingroup$ May I suggest to use a different phrasing than "net benefit". This is not a transaction where the benefits to the two sides are subtracted. $\endgroup$ May 5 at 6:25
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    $\begingroup$ good point, I changed "net" into "dominant". $\endgroup$ May 5 at 6:37
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    $\begingroup$ Physics related to observations, I don’t think string theory is there quite yet. $\endgroup$ May 5 at 13:16
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    $\begingroup$ For what it’s worth, much of the work Witten was awarded the Fields medal for was in quantum field theory and general relativity, not string theory. $\endgroup$ May 6 at 0:42
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    $\begingroup$ My impression was that Witten was mostly awarded the Fields Medal for his work on topological QFTs and low-dimensional geometry/topology, rather than string theory. $\endgroup$ May 7 at 13:09
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I think the best example for interactions between engineers and mathematicians is FEM —Finite Element Method— and FEA —Finite Element Analysis—.

The finite element method (FEM) for solving partial differential equations in 2 or 3 spatial variables, comes from the need to solve difficult elasticity and structural analysis problems in civil and aeronautical engineering in the '40s which inspired Alexander Hrennikoff, structural engineer, and Richard Courant, mathematician to develop FEM at its early stage.

FEM obtained its real taking off in the '60s and '70s by the work of J.H. Argyris (University of Stuttgart), R.W. Clough (UC Berkeley), O.C. Zienkiewicz and many more.

Distinct formulations share one essential feature: mesh discretization of a continuous domain into a set of discrete sub-domains, called elements. A finite element method is characterized by a variational formulation associated to the miniminization of an error, a discretization strategy to build such mesh, solution algorithms and optionally a post-processing scheme. The main advantage of FEM over other methods for solving PDEs is that it allows to model physical processes over very complex geometries.

Recently several mathematicians, Zlamal, Wachspress, and Ciarlet & Raviart, among others, have extended the analysis of FEM to include elements with curved sides.

Although FEM has a big impact on the Engineering side, it must be acknowledged that it has also made substantial contributions on Mathematical Methods for PDEs, Optimization and Algorithm Development, being an active area of research until today.

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    $\begingroup$ This doesn't seem to meet Michael Hardy's requirement that the interaction did not make a substantial contribution to engineering. FEM is extremely valuable to engineers. $\endgroup$ May 5 at 15:26
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    $\begingroup$ @TimothyChow : I've rephrased the question as follows: "What are the most edifying examples of that phenomenon in recent centuries, whereby applications to fields other than mathematics greatly benefitted mathematical research when mathematicians had expected to be only the benefactors of those other fields?" $\endgroup$ May 6 at 16:40
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    $\begingroup$ @MichaelHardy Thanks for clarifying the question. The limiting factor is that I'm having a hard time thinking of situations where "mathematicians had expected to be only the benefactors" (irrespective of who, if anyone, benefited from the interaction). That's a pretty restrictive proviso, and it seems hard to obtain definitive evidence, in any particular case, of what people's expectations were. $\endgroup$ May 6 at 23:43
  • $\begingroup$ @TimothyChow : Everybody knows that for centuries some mathematicians have been contemptuous of applications outside of mathematics. That would not continue long if it were genearally expected that pure mathematics would benefit from applications. $\endgroup$ May 7 at 1:55
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    $\begingroup$ @MichaelHardy Quite possibly a significant proportion of dismissiveness of applications is motivated by a skepticism that the applied field really benefits, not vice versa. $\endgroup$
    – Will Sawin
    May 8 at 18:09
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Not sure if this is what you had in mind, but experiments on the random packing of tetrahedral dice were able to achieve denser packings than had been constructed by mathematicians at the time. Both engineering and mathematics have played a role in this subject, but arguably mathematics has been more enriched than engineering.

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If Fourier analysis came from the study of heat flow, then, although the benefit to engineering and the sciences was immense, so was that to mathematical research.

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    $\begingroup$ I'm confused that your own example contradicts your requirement that the interaction not make a substantial contribution to engineering. $\endgroup$ May 5 at 15:28
  • $\begingroup$ @TimothyChow : I've rephrased the question as follows: "What are the most edifying examples of that phenomenon in recent centuries, whereby applications to fields other than mathematics greatly benefitted mathematical research when mathematicians had expected to be only the benefactors of those other fields?" $\endgroup$ May 6 at 16:41

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