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Throughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no practical application. Then a physicist or engineer comes along and finds a use for it.

I know that advancements have been developed for String Theory (maybe the only useful thing to come out of String Theory). But, in that vein, what are some of the most advanced or obscure mathematics that have real world, practical application to engineering, economics, computer science or such (especially if they are not well known)? And what branch of mathematics do they belong to?

Ben Grossmann
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Lou
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    That your cell phone uses a fractal as the antenna and that that is the optimal choice.\ – Moo Jun 29 '16 at 20:26
  • I am skeptical of your claim that string theory is a "practical application". But if you believe it is, then current research in topology, algebraic geometry, differential geometry, etc has ties with it. –  Jun 29 '16 at 20:32
  • Let me clarify on String Theory. I'm not saying String Theory is practical -- it really seems to be of no practical value. BUT, I've learned that some of the mathematics developed for it may have practical value outside of String Theory. – Lou Jun 29 '16 at 20:48
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    What is obscure mathematics ? – Dietrich Burde Jun 29 '16 at 20:51
  • @Moo The cell phone thing would make a good answer. Please consider submitting it as one. –  Jun 29 '16 at 21:04
  • @DietrichBurde, Somewhat subject to interpretation, but I would have considered Algebraic Topology and Group Theory to be obscure prior to discoveries for their practical uses. Most, if not all, applied mathematics started off as theory, so I'd say pure mathematics that only a small number of people know constitutes 'obscure'. I would distinguish obscure from advanced in that, in my view, advanced takes a well established branch of mathematics and extends it whereas obscure might be an entirely new branch of mathematics. – Lou Jun 29 '16 at 21:10
  • I vote for that crazy noneuclidean minkowski geometry stuff that they use to predict the relativistic time skew of GPS satellites. – MJD Jun 29 '16 at 21:22
  • First, the premise that "pure mathematicians strive to create mathematics with no practical application" is just a cutesy traditional joke or mythology: don't get sucked into believing it. Second, that mythical process of "physicist or engineer coming along and finding a use for it" is comparably bogus. People just don't operate in quite such a caricatured manner. Also, the idea that there is a fantasy Platonic world in which one could do math apart from the actual physical world is a bit of a stretch, considering that we reside here. Math is the human attempt to resolve puzzles and confusion. – paul garrett Jun 29 '16 at 21:43
  • ... of course, when academic math, like anything else, gets commodified, then its original purposes and its possible genuine on-going purposes can get lost in the hustle to make money, get grants, impress people, impress engineering students with the alleged dysfunction of mathematics, and so on. – paul garrett Jun 29 '16 at 21:44
  • I voted to reopen because even though the premises of the question are (the usual) stereotyping and defeatist mythology, I'd wager that many other sincere people would have the same question(s)... if only because the same sort of stereotype-promoting, caricaturizing, bad-scholar "professors" are to be easily found many places. ... sure, I know, it's the general human tendency to find a target for ridicule, which does create a feeling of bonding in the non-targeted group, etc. But, still, srsly, ppl, ... – paul garrett Jun 30 '16 at 02:46
  • An article I saw in Scientific American about how a topological result informally called the Hairy Ball Theorem was used to explain how certain conditions could result in heart fibrillation...2. Some solutions to the most efficient (densest) way to pack congruent non-overlapping hyperspheres in higher-dimensional Euclidean spaces have found application in designing error-correcting codes. 3. The use of Number Theory in cryptography, which is of no practical use without computers. 4. Hilbert Space theory was developed well before Quantum Mechanics, where it is indispensable.
    • – DanielWainfleet Jun 30 '16 at 07:14