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It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.

I am looking for mathematicians who did just that. An example would be Kurt Gödel's succesful incursion into general relativity.

I am not looking for people who found an application of their work into another field, but for people who, for some reason, had to educate themselves in a domain of which they were previously ignorant, and made some important discovery there.

Edit : I am interested in "modern" examples, let us say "post-Gauss", since it was arguably easier to switch disciplines in ancient times.

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closed as too broad by quid, R W, Chris Godsil, Bjørn Kjos-Hanssen, Yemon Choi Jul 7 at 2:30

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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I can think of many examples of mathematicians who started out in one area but spent most of their careers in another area (e.g. Grothendieck started out in functional analysis). Does this sort of thing count? –  Paul Siegel Jul 4 at 13:42
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I think it is unreasonably broad to include examples of mathematicians that switched subfields or work in several subfields at the same time. If criteria of importance and degree of remoteness are invoked this becomes pretty opinion based. –  quid Jul 4 at 17:01
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I think physics is too close to mathematics, there will be a large number of examples similar to Godel like e.g. David Hilbert, but also a large number of less well known mathematicians. –  Count Iblis Jul 4 at 18:48
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Riemann only had one paper on number theory, and it attracted some interest from the number theory community. –  Brian Rushton Jul 4 at 23:58
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I fear we are reaching the stage where superficial answers appear and get upvoted despite not fitting the question, just they because they involve famous names or famous stories. (But those who want the question re-opened should open a thread on meta.MO arguing for this) –  Yemon Choi Jul 7 at 2:32

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Jean Leray was in differential equations before WWII, but hid his applied mathematics background from his captors at a POW camp, where Leray did some fundamental work on spectral sequences that are widely used in algebraic topology/geometry and other fields today.

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Werner Nahm not only made important contributions to conformal field theory, but also conducted research about the Mayan civilization and their astronomy. In his Mayan research, he also worked with Linda Schele and Nikolai Grube and participated in the ongoing decipherment of Maya hieroglyphs.

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In this forum it might be appropriate to mention the contributions of user 766 to signal processing, which led to the discovery of the technique of compressed sensing, and a revolution in medical MRI.

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I think big problems in a field being solved by people outside the field in question is often the norm. See for instance the recent solution of the Kadison-Singer conjecture by Marcus-Spielman-Srivastava. The following quote from this AMS Bulletin article of Fulton on Horn's conjecture seems appropriate:

It may be worth pointing out explicitly that although the problems solved in this story range over several areas of mathematics — including linear algebra, commutative algebra, representation theory, intersection theory, and combinatorics — none of the people involved in the recent success came to the problems from any of these fields. Klyachko came from studying vector bundles on toric varieties; Totaro from studying filtered vector spaces using geometric invariant theory; Knutson, Agni- hotri and Woodward came from symplectic geometry; Tao from harmonic analysis; and Belkale from the study of local systems on Riemann surfaces.

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I don't think it's all that common - there is just sample bias as to how these things get reported. Also, not every important contribution is the solution of a big problem. –  Yemon Choi Jul 4 at 17:48
    
Also, from the OP: "I am not looking for people who found an application of their work into another field, but for people who, for some reason, had to educate themselves in a domain of which they were previously ignorant, and made some important discovery there." Marcus-Spielman-Srivastava does not fit this requirement -- the whole point is that other people had reduced KSP to something more combinatorial, and then MSS could bring their expertise to bear on it –  Yemon Choi Jul 4 at 17:51
    
Fair enough. But often when techniques from Field B get applied to major problems in Field A, those techniques become incorporated into Field A. For instance, I doubt anyone thought commutative algebra had much to do with combinatorics d poltroons before Richard Stanley's amazing proof of the Upper Bound Theorem, but now the two fields are seen as intimately linked. –  Sam Hopkins Jul 4 at 18:16
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@SamHopkins: poltroons?!? –  KConrad Jul 5 at 4:20
    
ah, that was my phone. "combinatorics of polytopes" –  Sam Hopkins Jul 5 at 14:00

Norbert Wiener got his PhD with a dissertation on logic, and worked in dynamical systems, ergodic theory, information theory... Still, he is probably most remembered for his work on engineering, communications, control systems and cybernetics.

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Probably by "their own field" you mean the whole of mathematics. There are plenty of mathematicians who work and make important contributions in several fields of mathematics, sometimes very remote from each other. But here are some outstanding examples of contributions outside of mathematics:

  1. Joseph Fourier (his main contributions to mathematics are "after Gauss", so I think he qualifies:-) Actually there are two broad fields where he is considered a "founding father", Egyptology and medical statistics. Specialists in Egyptology and medical statistics are even sometimes surprised when I tell them that his contribution to Math was also important:-) But in the city of Grenoble he is mostly remembered as an outstanding administrator, he was a prefect of the province, and organised many important improvements in his province.

The examples are really abundant. I am not even mentioning physicists who obtain Fields medals and mathematicians who obtain Nobel prizes in Economics, because everyone knows them. Here is a short list of people who come immediately to my mind. In XX century:

  1. Paul Painleve (after the Painleve equations, he switched to aviation and politics, and even was a prime minister for some time).

  2. Pierre Fatou. His was an astronomer by profession. He actually did many observations, and wrote important papers on celestial mechanics, computation of orbits and optics of the instruments. His seminal contributions to pure mathematics are well-known. Probably the last astronomer who was also a great pure mathematician.

  3. J. von Neumann. (Besides mathematics, computer science, economics, politics).

  4. Mstislav Keldysh. He and Lavrentjev were the fathers of the Soviet space program. Keldysh was the "Chief theorist" of the program. As a pure mathematician he is famous for deep results in Potential theory and Approximation theory.

  5. Michail Lavrentjev, one of the famous analysts of the first half of 20-s century, also contributed a lot to all sorts of applied sciences, and was an important politician. He created a city in Syberia where all kinds of scientific research is the main occupation of the inhabitants:-)

  6. Stanislaw Ulam (a pure mathematician with main interests in set theory, measure theory etc. He also had a patent for the first hydrogen bomb, jointly with E. Teller).

  7. Israel Gelfand. About 1/2 of his enormous production was in medicine.

  8. Donald Knuth. Mathematician who created TeX for all of us. You may dispute whether this is "inside" or "outside" Math:-)

Even among the people I know personally:

  1. Andrei Gabrielov. If you compare the list of his works on Google Scholar with the list on Mathscinet, you see that he made almost equally important contributions to pure mathematics and Geology.

  2. Alan Sokal. A famous mathematician and physicist. But most famous for his criticism of modern philosophers:-)

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+1 for Donald Knuth. –  André Henriques Jul 5 at 13:38
    
Heh, interesting. To any computer scientist, Knuth is first and foremost a computer scientist, probably the most legendary and immediately recognizable one currently alive. The development of TeX is just one of his many achievements in the field. –  JohannesD Jul 5 at 18:13
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But he also made significant contribution to pure math. Combinatorics, in particular. Look at Mathscinet, for example. –  Alexandre Eremenko Jul 5 at 18:26
    
@AlexandreEremenko: I'm sure of that; I just meant that from my perspective, he's more like a computer scientist who also made important contributions to math :) –  JohannesD Jul 10 at 7:38
    
And from my perspective, he is an outstanding mathematician who made a substantial contribution to Computer science. And invented TeX:-) –  Alexandre Eremenko Jul 10 at 14:29

Alan Turing, widely regarded as the father of Computer Science, is most widely-cited for his paper The Chemical Basis of Morphogenesis (wikipedia entry). The paper is integral in the development of theoretical biology and chaos theory.

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How important was Turing's proposed explanation of morphogenesis? –  Yemon Choi Jul 7 at 2:33

Abraham Robinson was interested in pure mathematics as an undergraduate, but became an expert in aerodynamics during World War II. After the war, he did his PhD thesis in model theory.

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Paul Cohen was an analyst but got a Fields Medal for his work in set theory, proving the independence of the continuum hypothesis from ZFC and the independence of the axiom of choice from ZF.

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Grassmann made his most important contribution in Linguistics.

Poincaré, besides his enormous work in mathematic, also add very important contribution to physics, and his work in philosophy of science was quite influential for a time, and recompensed by a nomination at the French Academy, quite a rare honor (?) for a scientific.

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Do you mean linguistics was Grassmann's most important contribution in his lifetime, or does it seem more important to linguistics today than his contributions in mathematics as they are now understood? –  KConrad Jul 5 at 4:32
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In his lifetime, as you know, his linguistic contributions were well-recognized while his mathematics contribution were not al all. With the benefit hindsight, well, it is difficult for me to say. In both cases, I think his contributions, while very valuable in themselves, are also impressive by their visionary quality, indicating very early a road that the two sciences were going to take : abstract linear algebra and algebraic geometry in math, strict phonetic transformation laws in historical linguistics. –  Joël Jul 5 at 12:18

Emmanuel Lasker leaps to mind: http://en.wikipedia.org/wiki/Emanuel_Lasker#Academic_activities_1894.E2.80.931918

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+1 just for leading me to find that Lasker has an entry in the MacTutor Biographies: www-groups.dcs.st-and.ac.uk/history/Biographies/Lasker.html –  Yemon Choi Jul 4 at 18:38
    
Thank you for letting me know Lasker, but I don't think he qualifies. If I understand well, he was a chess champion who was also educated in math and philosophy, not a mathematician who switched fields (within our outside math). –  Fabien Besnard Jul 4 at 19:44
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He thought of himself as a mathematician, has a doctorate (back when those meant something...), was employed (episodically) as a mathematician, and is the Lasker in the Lasker-Noether theorem, which is a fundamental result. The fact that he is more prominent as a chess player seems to be neither here nor there. –  Igor Rivin Jul 5 at 1:42

Personally I consider John Milnor and Steven Smale as two brilliant mathematicians who contribute in different areas. Milnor contributed to his work in differential topology, K-theory and dynamical systems and Smale contributed to topology and dynamical systems.

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Number-theorist G H Hardy is well-known to population biologists for the Hardy-Weinberg Law.

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As important as it may be in population biology, Hardy regarded the result as trivial and didn't have to learn any new math to make this contribution. –  KConrad Jul 5 at 4:16
    
@KCo, quite right, but the way the question is posed, what we need to know is whether Hardy "had to educate [himself] in a domain of which [he was] previously ignorant," that is, did he have to learn a little bit of biology? –  Gerry Myerson Jul 5 at 4:20
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The background leading to Hardy's involvement in this matter is at genetics.org/content/179/3/1143, where Punnett says "I put my problem to him as a mathematical one." This suggests to me that Hardy wouldn't have needed to learn something new to make his contribution. His paper briefly uses some terms from genetics, but if he didn't know those terms earlier than possibly he learned them from Punnett while writing the paper. –  KConrad Jul 5 at 4:56

David Mumford, a well known algebraic geometer, is responsible for the "Mumford-Shah segmentation" model in mathematical imaging. Besides being one of the most cited papers in this area, it also sparked in immense amount of work in geometric measure theory, "special functions of bounded variations", Gamma-convergent numerical approximations and calibration and functional lifting for numerical purposes. To get a glimpse, check the book Singular Sets of Minimizers for the Mumford-Shah Functional.

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I propose Jim Simons http://en.wikipedia.org/wiki/James_Harris_Simons

He was a great mathematician, then he became a "big" in economy.

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What do you mean "in economy"? Simons is not an economist. He is a successful money manager (or, more precisely, manager of a money manager), which is not a scientific pursuit. –  Igor Rivin Jul 5 at 1:44
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Are any of his important discoveries in quantitative finance (assuming they exist) published anywhere? –  S. Carnahan Jul 5 at 2:24

If I'm not mistaken, Leonid Kantorovich started as a mathematician then became known as an economist. (I'm not too familiar with his work; perhaps someone else can edit this answer to give more detail.)

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Kantorovich's work is in mathematics that is applicable to economics. I don't think he ever thought of himself as an economist. –  Igor Rivin Jul 6 at 0:57

Thurston (together with Sleator and Tarjan) solved a problem in combinatorics about "rotation" distance between binary trees. The solution translated the problem into a question in hyperbolic geometry, but didn't directly make use of Thurston's previous work in that field.

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It did not make use of actual results, but notice that Thurston's best known results are precisely like this one: they translate combinatorial questions (e.g., classification of three-dimensional manifolds) into geometric questions, where one can then make more progress. So, I would argue that Sleator/Tarjan/Thurston is not outside Thurston's field at all. –  Igor Rivin Jul 6 at 0:59

Leonard Adleman (as in RSA Algorithm) made significant contributions to AIDS research.

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This is a magnificent example, thank you. –  Fabien Besnard Jul 5 at 9:30

I think that Robert Solovay ought to be mentioned here for his contributions to prime numbers, namely the Solovay–Strassen primality test.

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Also the Solovay-Kitaev theorem, about density of finite subgroups of matrix groups. It was a significant step in the theory of quantum computing. –  Andreas Blass Jul 7 at 16:14

I think Ronald Fisher's Ph.D. was in mathtematics and he published some things on differential geometry.

He is one of the three major founders of the science of population genetics, with Wright and Haldane. His writings on biology are fairly voluminous.

He is also the originator of many of the things taught in basic theory-of-statistics courses, including sufficiency and Fisher information. He single-handedly founded the discipline of design of experiments.

He introduced fiducial inference in order to apply it to what became known as the Behrens–Fisher problem. It's often hard to tell just what Fisher intended in things he wrote. Someone named Bartlett in 1936 published a proof that Fisher's fiducial intervals don't have constant coverage rates. Constant coverage rates are part of the definition of confidence intervals. Bartlett seemed to suggest that it is therefore an error to use fiducial inference. Fisher replied that he never intended his intervals to have constant coverage rates, but I haven't been able to figure out just what he did intend. Bayesian credible intervals also don't have constant coverage rates, but in that case everyone understands why that should be so. If I knew a bit more than I do about the Behrens–Fisher problem, I'd an write expository paper on why it is not and cannot be really a math problem and any attempt to pretend it is one is a misunderstanding.

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Could you say more about what you know about Fisher's work in differential geometry? I just checked his publications on Mathscinet and found no evidence of this. On the other hand, the earliest publication listed is in 1939 and, according to the Genealogy Project, Fisher got his doctorate in 1926. –  Deane Yang Jul 5 at 16:44
    
I was about to say that unfortunately I "read somewhere" that this was the case. What I've got now is this: msh.revues.org/3631?file=1 –  Michael Hardy Jul 5 at 18:25
    
"Neither advanced their subjects in important ways, but the second showed a firm grasp of an astonishing range of modern mathematics." Grammatically grating. I'd have written "Neither advanced its subject in an important way". –  Michael Hardy Jul 5 at 18:26
    
I was surprised to see the date of his degree was 1926, since he was publishing a lot of stuff long before that. –  Michael Hardy Jul 5 at 18:27
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According to the article you linked to, Fisher wrote only an expository essay on differential geometry. That doesn't seem like much to me. Fisher was a great statistician, but I don't think he did much outside his field. I confess to being disappointed. There definite connections between what is known as Fisher information and differential geometry. It would be nice to see that Fisher somehow knew this from the start. –  Deane Yang Jul 5 at 20:14

You know, a good example in the other direction is John Banzhaf and the Banzhaf index. He studied engineering as an undergraduate and is a practicing attorney, but made a major, albeit anticipated, contribution to Game Theory. (Until checking Wikipedia, I'd never known it was anticipated, and I suspect the original discovery went un-noticed.)

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John Tukey's Ph.D. is Mathematics (from Princeton, in Topology). His contributions are in applied statistics and computation. Tukey was responsible (in part) for the FFT.

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(+1) But, with the slightly ironic caveat that the OP specifies "post-Gauss", yet Gauss knew the FFT. Tukey's contributions to statistics and allied areas of engineering went beyond that, of course. –  cardinal Jul 6 at 17:50
    
The FFT isn't the reason Tukey belongs on the list, but I thought it more likely to be recognized than, say, robust estimation or EDA. –  Dennis Jul 7 at 2:19

I think of John Nash. Basically, he is mathematician, specielized in differential geometry, number theory, game theory. And he brought such huge discoveries in economics that he got the Nobel Prize of Economy.

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-1. Quoting the OP: " am not looking for people who found an application of their work into another field, but for people who, for some reason, had to educate themselves in a domain of which they were previously ignorant, and made some important discovery there." –  Yemon Choi Jul 6 at 19:14
    
The motivation given in the answer is perhaps not completely in line with OP, but then I fail to see why Nash should be a less legitimate answer than, say, Milnor or numerous other answers. –  quid Jul 6 at 19:26
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@quid I've just seen, and downvoted, the Milnor answer –  Yemon Choi Jul 7 at 2:29

I'd mention Ion Barbu, one of the greatest Romanian mathematicians and also one of the greatest Romanian poets. To the trained reader, the structure and feeling of his poems betray his mathematical mindset, a mindset that served in the making of his own lyrical style, which is unique in Romanian literature.

As far as I know, his greatest contribution in the field of mathematics revolves around Barbilian spaces, which bear his name.

Quoting wikipedia:

At the University of Missouri in 1938 L.M. Blumenthal wrote Distance Geometry. A Study of the Development of Abstract Metrics, where he used the term "Barbilian spaces" for metric spaces based on Barbilian's function to obtain their metric.

Wiki for Ion Barbu

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In his early years, Andrey Kolmogorov was interested in Russian history. Citing Scholarpedia:

He did serious scientific research on XV-XVI century manuscripts concerning agrarian relations in ancient Novgorod. In the twenties he made a hypothesis on the way the upper Pinega was settled, and this conjecture was later confirmed by an expedition to that area.

It is said that he totally moved to mathematics when explained that in history, five different proofs are required for each assertion. There is a related discussion on History Stack Exchange.

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James Garfield, in addition to being a published mathematician was also a former President of the United States. He reformed the civil service system and worked on civil rights issues.

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The order of accomplishments is backwards, kind of like saying "Thomas Edison, in addition to being a father, was also an inventor". –  KConrad Jul 7 at 2:33

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