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It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:

What are some applications of other fields to mathematics?

Obviously the applications of physics to mathematics are ubiquitous (gauge theory is just one significant modern example, and quantum algorithms and mirror symmetry are others...the list from physics goes on). For the purposes of this question (at least) theoretical computer science is just a branch of mathematics.

So answers involving fields other than physics are of particular interest to me (and answers involving theoretical computer science are of little to no interest to me), as are answers where the application isn't bidirectional (for example, one could say that game theory is an application of mathematics to economics as much if not more than an application of economics to mathematics).

Finally (at least for the purposes of this question), anything of the form "phenomenon Y was experimentally observed and it turned out that there was a rich but hitherto unknown mathematical theory Z explaining Y" is not that interesting as an application to mathematics unless the discovery of Z has some truly special status. Something like (e.g.) symplectic geometry might fall under this (leaving aside the "experimental" bit), but is not of particular interest for reasons above.

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I have deleted a discussion in the comments about having too many "big-list" questions on the front page. Please follow the link above to meta if you're interested in that discussion. –  Scott Morrison Feb 9 '10 at 18:49
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18 Answers

Here's a nice paper by Sturmfels, on the question Can biology lead to new theorems?

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Kevin--This is a great answer, exactly the sort of thing I was looking for. Thanks! –  Steve Huntsman Feb 10 '10 at 3:44
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According to Gordan, the Hilbert Basis Theorem was an application of theology.

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You don't understand either Gordan's comment or theology, or this is a facetious answer to a serious, even if somewhat broad, question. –  Victor Protsak Aug 14 '10 at 7:44
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@Victor, one man's facetious is another man's mildly amusing. I'm sorry that my sense of humor does not align with yours. –  Gerry Myerson Aug 14 '10 at 23:05
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I can describe one that is still a little mysterious to me. My colleagues Erwin Lutwak and Gaoyong Zhang and I have shown how ideas arising from the continuous version of Shannon information theory (which normally resides in the electrical engineering department) lead very naturally to sharp analytic inequalities for functions on $R^n$, including generalized sharp Sobolev inequalities. What's really nice is that not only does this point of view lead to the inequalities, it also leads to much nicer and easier to understand proofs of the inequalities than previously known proofs.

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It is perhaps a bit of a historical fluke that information theory lives in EE. This, of course, stems from its roots in communication theory. Pure information theory might be better characterized as a branch of statistics, or possibly even physics. That said, another good example along these lines is the paper Information theoretic inequalities by Dembo, Cover & Thomas, where, for instance, some classical determinantal inequalities are derived from consideration of entropies of multivariate normals. –  Jon Yard Feb 10 '10 at 1:18
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@Jon: You make a good point, and I really should have cited the work of Dembo, Cover, and Thomas, as well as other people. They have done some beautiful mathematics disguised as EE, and I think there is a lot more to be explored. –  Deane Yang Feb 10 '10 at 1:56
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@Jon Yard, @Deane Yang, wasn't there a turf battle between mathematics and electrical engineering for "ownership" or "rights" to computer science and information theory. At some places, computer science is a part or division of Electrical Engineering. At few, computer science is a branch of mathematics. And at many places, it is a department in its own right. The same thing happens with Neurosurgery. Some schools have Divisions of Neurosurgery as part of the Department of Surgery, others free-standing Depts of Neurosurgery, and for some odd ducks it's part of Neurology, a medical branch. –  Apurva Sep 16 '10 at 22:19
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There is a lot of evidence that many EE and CS professors are really mathematicians, even pure mathematicians, who figured out that life would be much better (higher salaries, lower teaching load, better grants) if they disguised themselves as working in EE or CS. There is a lot of extremely deep mathematics being done in the name of EE or CS. –  Deane Yang Sep 17 '10 at 1:58
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I can think of at least three things that the question might mean, and it would probably help if Steve clarified which ones count for him!

(1) Other fields suggesting new questions for mathematicians to think about, or new conjectures for them to prove. Examples of that sort are ubiquitous, and account for a significant fraction of all of mathematics! (Archimedes, Newton, and Gauss all looked to physics for inspiration; many of the 20th-century greats looked to biology, economics, computer science, etc. Even for those mathematicians who take pride in taking as little inspiration as possible from the physical world, it's arguable how well they succeed at it.)

(2) Other fields helping the process of mathematical research. Computers are an obvious example, but I gather that this sort of application isn't what Steve has in mind.

(3) Other fields leading to new or better proofs, for theorems that mathematicians care about even independently of the other fields. This seems to me like the most interesting interpretation. But it raises an obvious question: if a field is leading to new proofs of important theorems, why shouldn't we call that field mathematics? One way out of this definitional morass is the following: normally, one thinks of mathematics as arranged in a tree, with logic and set theory at the root, "applied" fields like information theory or mathematical physics at the leaves, and everything else (algebra, analysis, geometry, topology) as trunks or branches. Definitions and results from the lower levels get used at the higher levels, but not vice versa. From this perspective, what the question is really asking for is examples of "unexpected inversions," where ideas from higher in the tree (and specifically, from the "applied" leaves) are used to prove theorems lower in the tree.

Such inversions certainly exist, and lots of people probably have favorite examples of them --- so it does seem like great fodder for a "big list" question. At the risk of violating Steve's "no theoretical computer science" rule, here are some of my personal favorites:

(i) Grover's quantum search algorithm immediately implies that Markov's inequality, that

$\max_{x \in [-1,1]} |p'(x)| \leq d^2 \max_{x \in [-1,1]} |p(x)|$

for all degree-d real polynomials p, is tight.

(ii) Kolmogorov complexity is often useful for proving statements that have nothing to do with Turing machines or computability.

(iii) The quantum-mechanical rules for identical bosons immediately imply that |Per(U)|≤1 for every unitary matrix U.

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I like your (i), though I'm accustomed to what looks like a different Markov inequality: en.wikipedia.org/wiki/Markov%27s_inequality ...also Kevin's answer gives a very good example of the sort of thing I was looking for. –  Steve Huntsman Jul 25 '10 at 17:18
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@Scott, your apt biological similes "leaves, roots" (etc.) are realized (fairly literally) in Misha Gromov's recent Bull AMS review "Crystals, proteins, stability and isoperimetry", passages from which I have quoted below: 'mathoverflow.net/questions/14782/…; –  John Sidles May 31 '11 at 22:33
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There's a connection between random walks and electric networks (the link goes to the book of that title by Doyle and Snell); this is "physics", I suppose, but hopefully not the sort you meant to exclude!

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This is pretty good for an application from physics (or EE). I actually mentioned this book in an answer to another question. –  Steve Huntsman Feb 10 '10 at 3:44
    
I thought I'd seen it mentioned before on MO. –  Michael Lugo Feb 10 '10 at 3:56
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This blog post of Kenny Easwaran suggests a mathematical result that can be explained using economic intuition. It's more an interesting isolated example than a genuine application, but it's still interesting.

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Gerry Myerson's answer about Gordan and theology was humorous, but Georg Cantor really did use theology in his conception of set theory. From Cantor's Wikipedia biography:

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[51] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[52] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God,[53] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[12]

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One of the first examples of Markov chains came from A.A. Markov's study of vowel/consonant succession in Pushkin's "Eugene Onegin":

An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains, trans. David Link. Science in Context 19.4 (2006): 591–600. Online: http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=637500

This is more of an application of poetry to mathematics (in order to give an example illustrating a new mathematical concept) than the other way round.

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Not sure if this counts, but two rather effective global optimization algorithms of stochastic type were in fact inspired from work in different fields: "simulated annealing" rests on a physical analogy to the slow cooling (annealing) of metals, while "differential evolution" appeals to an analogy to mating of pairs and the subsequent mutation of their offspring.

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Misha Gromov's recent Bull. AMS article "Crystals, proteins, stability and isoperimetry" (2011) can be read as a 29-page essay on the requested topic, with a focus particularly on mathematical inspiration arising in evolutionary biology, neurophysiology, and cognitive science. Gromov sets the stage as follows:

One may conjecture that neither cell nor brain would be possible if not for profound mathematical “somethings” behind these, Nature’s inventions. But what are these “somethings”? Why do we, mathematicians, remain unaware of them? … The history of mathematics shows how slow we are when it comes to inventing/recognizing new structures even if they are spread before our eyes, such as hyperbolic space, for instance. … One has to browse through myriad stars—structural specks of Life revealed by biologists—in order to identify the “essential ones”, and when (if ?) we find them, we may start on the long road toward new mathematics.
Gromov then goes on to suggest many dozens of concrete questions, arising in many bio-related disciplines, that uniformly direct our vision (to use Scott Aaron's nice similes) from the "leaves" of life to the "roots" of fundamental mathematics.

What does Gromov see (that everyone sees) that inspires him so frequently to conceive mathematics (that no one previously has conceived (Szent-Gyrgyi))? Gromov has written this essay, to tell us precisely what it is, that he presently sees.

Since this is a community Wiki, I will informally suggest that it is great fun to read Gromov's inspiring essay either immediately before, or immediately after, viewing Stephane Guisard and Jose Salgado's similarly inspiring VLT (Very Large Telescope) HD Timelapse Footage.

Gromov is concerned largely with very small (molecular-scale) evolved systems, while Guisard and Salgado are concerned mainly with very large (galactic scale) evolved systems … and yet they are tapping the same source.

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The Lotka-Volterra predator-prey equations are a fundamental example in the qualitative theory of ODEs. Volterra originally used it to explain the large increase in the mediterranean shark population during WWI.

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As a contribution to mathematical practice (as opposed to mathematics itself), you might consider the emerging field of mathematical cognition (also known as cognitive science of mathematics), i.e. the study of mathematical ideas and their empirical grounding in human experiences, metaphors, generalizations, analogies and other cognitive mechanisms.

This subject has been explored informally and non-rigorously by mathematicians such as Saunders Mac Lane (see his Mathematics, Form and Function), but until recently it was not pursued by researchers trained in cognitive science. The best-known introduction is the book Where Mathematics Comes From by G. Lakoff and R. Nunez, but research has also been undertaken by Brian Rotman (Mathematics as sign: Writing, imagining, counting) and others.

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Chaitin describes in his book "Meta Math! The Quest for Omega" his point of view on information theory, complexity theory and a number of other questions (some of which should not be taken too serious, e.g. when he's comparing evolution and quantum physics). The whole book is dedicated to telling the story of how he discovered whta is now called Chaitin's constant and the theory connect with it by thinking about rather simple(looking) questions in computer sciences. (It is therefore a very unusual math book)

If someone is interested in how non-math-question can lead to new math, this book is definitely one of the places to find examples. You can find a online reader for the book here.

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Rodney Baxter was led to rediscover the Rogers-Ramanujan identities in the course of his work on the hard hexagon model of a gas:

http://en.wikipedia.org/wiki/Hard_hexagon_model

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There have been many theoretical advances in diffusion, traveling waves, chaos, and pattern formation derived from the Belousov–Zhabotinsky reaction.

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I’d like to add something regarding game theory. Certainly, it is generally true that – as stated in the question - the application isn’t bidirectional.

However, there are exceptions. I heard of some contributions of the so called “Infinite games with perfect information” to the fields of:

mathematical logic and set theory – e.g. the Axiom of determinacy,

topology – topological games.

I think that especially the contribution to mathematical logic is a nice example of the ”unexpected inversion” (a term used by Scott Aaronson in his answer to this question). We have a situation when the applied leave (game theory) contributes to the root of the tree.

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One should perhaps also mention phyllotaxis.

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Should one perhaps also mention what it is? –  Nate Eldredge May 22 '13 at 12:56
    
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Mathematics need some source of new ideas, and any other fields are very good sources of ideas: art, philosophy, physics, music, even sport probably. From the other side may important discoveries are made by means not-so-precise formulations of solutions known problems. Feynman path integral is good example, distributions theory as well etc. Forgive me, but math without such external source of ideas, without any contact with reality ( not exactly direct contact) in my opinion is not very productive nor interesting.

In fact it shares this property with any other kind of art.

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