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Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant interests that one wouldn’t guess one from the other?

(Best if the two interests are not endpoints of a continuum, as may have been the case of past universalists like Euler or Cauchy. For this reason, maybe best restrict to post-1850 or so?)

The point of asking is that it seems not so rare, but you don’t normally learn it other than by chance.


Edit: Now CW, works best with “one mathematician per answer” (and details of actual achievement, e.g. “war work on radar” may have been creative for some but maybe not all who did it).$\,\!$

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    $\begingroup$ Almost all known mathematicians are also known for their applied achievements, especially the older mathematicians. For instance, topologist Karol Borsuk, when living under the WWII German occupation, had developed an entertaining game from sales of which he derived his income which helped him to survive in those harsh times. In later years, algebraic topologists, the first students of S.Novikov, namely Vitia Bukhshtaber and Sasha Mishchenko, worked intensively on applied projects. The list of examples is endless. $\endgroup$ – Wlod AA Jan 2 at 0:27
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    $\begingroup$ I asked exactly the same question elsewhere and got shut down with almost no answers. mathoverflow.net/questions/345636/… $\endgroup$ – Hollis Williams Jan 2 at 19:35
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    $\begingroup$ @Tom, that' MO-life/game, it comes with the territory. $\endgroup$ – Wlod AA Jan 2 at 20:17
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    $\begingroup$ @Wold AA, haha, no, that's the way some very highly-opinionated, demonstrative people get their jollies when dealing with outsiders, as you find in most facets of life. Don't condone it by normalizing it. $\endgroup$ – Tom Copeland Jan 2 at 23:57
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    $\begingroup$ Possible duplicate of Examples of Mathematicians who excelled in Pure and Applied Mathematics. (I apologize for answering this question and not the earlier one. I was not aware of the earlier one.) $\endgroup$ – Joel Reyes Noche Jan 3 at 2:12

62 Answers 62

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John von Neumann was the first person to come to my mind.

He published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones.

[…]

In a short list of facts about his life he submitted to the National Academy of Sciences, he stated, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

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    $\begingroup$ Not to forget his computer architecture with a single central processing unit $\endgroup$ – Manfred Weis Jan 1 at 7:44
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    $\begingroup$ And at the opposite extreme, his contributions to set theory --- you can't get more pure than that. The modern definition of an ordinal as the set of all smaller ordinals is due to him. (I was told the paper where he did this was written when he was in high school.) $\endgroup$ – Nik Weaver Jan 1 at 18:34
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    $\begingroup$ In principle, it was John vin Neumann who once and forever, up to details, defined life, and, following Ulam's additional ideas, constructed first artificial living "organism". His notion of life was somewhat narrow which made sense as the crucial starting point. $\endgroup$ – Wlod AA Jan 1 at 23:56
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    $\begingroup$ Also the first person to define the concept of a computer virus, now ubiquitous. $\endgroup$ – Hollis Williams Jan 2 at 19:36
  • $\begingroup$ I realise that Wlod was referring to the computer virus when he talked about the construction of an artificial living organism. $\endgroup$ – Hollis Williams Jan 2 at 23:37
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Alan Turing is one mathematicians with both “very abstract” and “very applied” achievements: https://en.wikipedia.org/wiki/Alan_Turing

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    $\begingroup$ His applied work is diverse: He broke nazi codes and he worked on the chemical basis of morphogenesis, predicting oscillating chemical reactions before they were observed. $\endgroup$ – Michael Hardy Jan 1 at 16:16
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    $\begingroup$ @MichaelHardy, "He [Turing] broke nazi codes" -- Turing did no such thing, but, yes, he was a part of a huge team working on breaking Enigma (German encrypting machine). It has started with Polish mathematicians. British mathematicians/chessplayers were later in charge. They enrolled in a future topologist (homotopy), young -- at the time -- Peter Hilton. etc. $\endgroup$ – Wlod AA Jan 1 at 23:50
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    $\begingroup$ Would it be possible to expand this answer? E.g. giving examples of his abstract and applied achievements. $\endgroup$ – Rebecca J. Stones Jan 2 at 8:18
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Jean Leray started in fluid mechanics. His work might not be applied enough for your question, but when he was made prisoner during WWII he concealed his knowledge about fluid mechanics thinking he could be asked to provide expertise to the enemy. Instead he developed sheaf theory and spectral sequences while a prisoner.

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  • $\begingroup$ Interesting! Thank you. $\endgroup$ – Wlod AA Jan 1 at 23:53
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    $\begingroup$ So in a sense, sheaf theory is "pacifistic" mathematics. I like that. $\endgroup$ – Martin Brandenburg Jan 3 at 12:44
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David Mumford did foundational work in algebraic geometry, but then later in his career did a lot of work in the applied areas of vision (especially computer vision) and pattern theory.

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  • $\begingroup$ Did any of the computer vision work pan out (no pun intended)? $\endgroup$ – Tri Jan 2 at 6:56
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    $\begingroup$ @Tri: not an expert so can't say for sure, but my impression is that yes his work in vision was also fundamental, e.g., see en.wikipedia.org/wiki/Mumford%E2%80%93Shah_functional which apparently is actually used in modern image programs. $\endgroup$ – Sam Hopkins Jan 2 at 14:46
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Has everyone forgotten John Nash? I believe his contributions to game theory are among the most prominent examples of mathematical ideas which are widely used in other fields.

On the pure end, the De Giorgi-Nash(-Moser) theorem is a landmark of elliptic and parabolic PDE, which establishes the uniform control of solutions of linear PDE with no assumptions on the smoothness of the coefficients - this is widely used in applications to the nonlinear case. Nash also resolved the isometric embedding problem in differential geometry with a very clever analytic approach. The main part of his proof established a particular instance of what is now known as the Nash-Moser implicit function theorem. The statement is somewhat innocuous (following Richard Hamilton's formulation, it extends the implicit function theorem from Banach spaces to 'tame Frechet spaces') but Nash's proof was very daring. Nash also resolved the isometric embedding problem in a different way, by looking for a low-regularity solution. The 'impossible'-looking thing about this paper is that he shows that n(n+1)/2 simultaneous PDE can be satisfied by finding only n+2 different functions, the key being that this is impossible if the n+2 functions are even as much as twice-differentiable. Nash's proof is remarkably direct.

(I remember, when he got the Nobel Prize in 1994, sitting at a lunch table with a group of mathematicians in different fields, each of whom knew of Nash from his work in their subject, all trying to figure out whether "their" Nash was the guy getting the prize.)

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  • $\begingroup$ "On the pure end ... do I need to list examples?" Yes, please! $\endgroup$ – Nathaniel Jan 3 at 10:59
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    $\begingroup$ @Nathaniel The De Giorgi-Nash(-Moser) theorem is a landmark of elliptic and parabolic PDE, which establishes the uniform control of solutions of linear PDE with no assumptions on the smoothness of the coefficients - this is widely used in applications to the nonlinear case. $\endgroup$ – slcvtq Jan 3 at 13:49
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    $\begingroup$ @Nathaniel Nash also resolved the isometric embedding problem in differential geometry with a very clever analytic approach. The main part of his proof established a particular instance of what is now known as the Nash-Moser implicit function theorem. The statement is somewhat innocuous (following Richard Hamilton's formulation, it extends the implicit function theorem from Banach spaces to 'tame Frechet spaces') but Nash's proof was very daring - I think that to most analysts it 'looks' like it has no chance of working. $\endgroup$ – slcvtq Jan 3 at 13:56
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    $\begingroup$ @Nathaniel and Nash also resolved the isometric embedding problem in a different way, by looking for a low-regularity solution. The 'impossible'-looking thing about this paper is that he shows that n(n+1)/2 simultaneous PDE can be satisfied by finding only n+2 different functions, the key being that this is impossible if the n+2 functions are even as much as twice-differentiable. Nash's proof is remarkably direct. $\endgroup$ – slcvtq Jan 3 at 14:00
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    $\begingroup$ @Nathaniel He also has a significant paper where he shows short-time existence for solutions of a fluid problem, although I'm unfamiliar with any of the details. In algebraic geometry, he showed that any manifold type can be realized as a real affine algebraic variety. I'm not sure how significant this result is, although it's well-known. $\endgroup$ – slcvtq Jan 3 at 14:03
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Stanisław Ulam is best known for his work on the Manhattan Project, but his contributions to pure mathematics include pioneering work on measurable cardinals and formulating the reconstruction conjecture in graph theory.

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Abraham Robinson made contributions to mathematical logic (model theory, nonstandard analysis) and aerodynamics (airplane wing design).

In December of 1942 Robinson wrote to his supervisors in Jerusalem that he had decided to participate in the general struggle against Fascism and apply his knowledge in applied mathematics to this end. He remarked that there was no effort for him to turn to applied problems. Robinson addressed the problem of comparison between single-engine and twin-engine planes for which he suggested an analog of the variational method by Ludwig Prandtl. He also worked on the problem of structural fatigue and collapse of a flying boat.

In 1944 Robinson married Renée Kopel, a fashion photographer. Abby lived with Renée up to his terminal day.

Robinson was a member of the group studying the German V-2 missiles as well as of a mission of the British Intelligence Objectives Subcommission which concerned intelligence gathering about the aerodynamical research in Germany. In 1946 Robinson returned to Jerusalem to pass examinations for the Master degree. The results were as follows: “physics good, mathematics excellent.” In this short period Abby worked together with Theodore Motzkin.

In 1946 the Royal College of Aeronautics was founded in Cranfield near London. Robinson was offered the position of a Senior Lecturer with salary 700 pounds per year. It is worth mentioning that Robinson was the only member of the teaching staff who learned how to pilot a plane. In Cranfield Abby became a coauthor of delta-wing theory for supersonic flights, and in 1947 he learned Russian in order to read the Soviet scientific periodicals.

To gain the PhD degree, Robinson joined the Birkbeck College which was intended for mature working students and provided instructions mainly in the evening or on weekends. Abby's supervisor in the college was Paul Dienes, a Hungarian specialized mainly in function theory. Dienes instigated Abby's interest in summation methods (which resulted lately in Abby's work with Richard Cooke who also taught in the Birkbeck College). Dienes was a broad-minded scientist with interests in algebra and foundations. In 1938 he published the book Logic of Algebra, the topic was close to Abby's train of thoughts. In this background Robinson returned to logic and presented and maintained the PhD thesis “On the Metamathematics of Algebra” in 1947.

In 1951 Robinson moved to Canada where he worked at the Department of Applied Mathematics of Toronto University. He delivered lectures on differential equations, fluid mechanics, and aerodynamics. He also supervises postgraduate students in applied mathematics. Abby worked on similarity analysis and wrote “Foundations of Dimensional Analysis” which was published only after his death in 1974.

Robinson was the theorist of delta-wing, but his Farnborough research in the area was highly classified. In Toronto Robinson wrote his magna opus in aerodynamics, Wing Theory, which was based on the courses he delivered in Cranfield as well as on his research in Canada. Robinson invited as a coauthor John Laurmann, his former student in Cranfield. The book addressed airfoil design of wings under subsonic and supersonic speeds in steady and unsteady flow conditions. James Lighthill, the creator of aeroacoustics and one of the most prominent mechanists of the twentieth century, appraised most of the book as “an admirable compendium of the mathematical theories of the aerodynamics of airfoils and wings.” Robinson performed some impressive studies of aircraft icing and waves in elastic media, but in the mid-1950s his interest in applied topics had been fading. Robinson continued lecturing on applied mathematics, but arranged a seminar of logic for a small group of curious students.

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    $\begingroup$ What was Robinson’s achievement in aerodynamics? The online information on this seems thin — did his wing designs get used, become influential? $\endgroup$ – Matt F. Jan 1 at 16:37
  • $\begingroup$ My impression was that Robinson's achievements in applied math are much less influential than his pure math work. His work on applied math is barely mentioned in the obituary which I read of him, although that might just be because of lack of knowledge on the part of the author. $\endgroup$ – Hollis Williams Jun 2 at 16:57
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G. H. Hardy is well-known for his work in number theory, but also for the so called Hardy–Weinberg principle in genomics.

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Some things Kolmogorov did were fairly "pure", but he is one of the eponyms of the Kolmogorov–Smirnov test.

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    $\begingroup$ And also contributed to the theory of turbulence in fluid mechanics. $\endgroup$ – Piyush Grover Jan 1 at 23:56
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    $\begingroup$ Actually, Kolmogorov is one of the finest examples! An extraordinary span of crucial results in pure mathematics but also for applications. $\endgroup$ – Wlod AA Jan 2 at 0:06
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    $\begingroup$ The modern approach to the theory of probability, based on the set theory and measure theory is due to Andrey Kolmogorov, he is the father of the modern probability theory. $\endgroup$ – Wlod AA Jan 2 at 20:26
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Steve Vickers works on topos theory and pointfree topology, but also

He was responsible for the adaptation of the 4K ZX80 ROM into the 8K ROM used in the ZX81 and also wrote the ZX81 manual. He then wrote most of the ZX Spectrum ROM, and assisted with the user documentation.

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Pontryagin is considered one of the best both pure and applied mathematicians in the last сentury.

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    $\begingroup$ @IgorRivin: everything he did after 1950 (approximately). Pontryagin max principle for example. Optimal control and so on. $\endgroup$ – user6976 Jan 1 at 20:01
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    $\begingroup$ Interesting! I had no idea... $\endgroup$ – Igor Rivin Jan 1 at 20:03
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    $\begingroup$ And, unfortunately, when I think of his name, one word dominates all others, and it is not "optimal control". $\endgroup$ – Igor Rivin Jan 1 at 20:04
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    $\begingroup$ He was very mentally sick at the end of his life (I think it is the result of lots of hard calculations in applied math). But what he did in mathematics is hard to overestimate. $\endgroup$ – user6976 Jan 1 at 20:06
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    $\begingroup$ @MarkSapir: I should hope most people who do hard calculations don't suffer from mental illness! $\endgroup$ – AlexArvanitakis Jan 2 at 16:12
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The first person who comes to mind is Hilbert, although his work on quantum mechanics might not qualify as „very applied“. But Wiener and von Neumann certainly fit the bill. Many of the prominent pure mathematicians in the UK (and presumably elsewhere) were involved in war work and did some very applied stuff there—a prominent example would be Robert Rankin. Gröbner in Austria is an analogous case.

Added as an edit: the fact that many pure mathematicians have worked as code breakers is mentioned elsewhere but of course Bletchley Park is a rich source of examples. Turing‘s contribution there is common knowledge but he was one of several, including, notably, William Tutte and Peter Hilton.

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    $\begingroup$ Hilbert also came up with the Einstein-Hilbert action before Einstein ... $\endgroup$ – Nik Weaver Jan 1 at 2:08
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    $\begingroup$ Einstein never came up with the Einstein-Hilbert action at all, he derived the Einstein equations using a geometric argument and never originally thought of obtaining them by varying an action. $\endgroup$ – Hollis Williams Jan 2 at 19:34
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Several known for “pure” work have rather applied contributions to geometrical optics:

  • Carathéodory (1937) Geometrische Optik
  • Chaundy (1919) The aberrations of a symmetrical optical system
  • Whittaker (1907) The theory of optical instruments
  • Hausdorff (1896) Infinitesimale Abbildungen der Optik
  • Hensel (1888) Theorie der unendlich dünnen Strahlenbündel
  • Kummer (1861) Über atmosphärische Strahlenbrechung
  • Weierstrass (1856) Zur Dioptrik
  • Sturm (1845) Mémoire sur la théorie de la vision
  • Listing (1845) Beitrag zur physiologischen Optik
  • Gauss (1843) Dioptrische Untersuchungen
  • Liouville (1842) Démonstration d’un théorème de M. Biot sur les réfractions astronomiques
  • Möbius (1830) Kurze Darstellung der Haupt-Eigenschaften eines Systems von Linsengläsern
  • Monge (1798) Mémoire sur le phénomène d’Optique, connu sous le nom de Mirage
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I am surprised that no one has mentioned Terence Tao yet. Speaking of his very abstract achievements in math to the audience of this site is like carrying coals to Newcastle. But he has some cool applied achievements as well. Such as compressed sensing, helping physicists model neutrino oscillations and the Polymath project.

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    $\begingroup$ Terence Tao is obviously an excellent mathematician, but he didn't model neutrino oscillations. He explained some math behind work of physicists who were modeling neutrino oscillations. $\endgroup$ – Jeff Harvey Jan 2 at 3:31
  • $\begingroup$ He did nothing to model neutrino oscillations, but he did explain some of the linear algebra which neutrino physicists were seeing in their work. $\endgroup$ – Hollis Williams Jun 2 at 16:35
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Israel Gelfand ... was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis ...

The Gelfand–Tsetlin (also spelled Zetlin) basis is a widely used tool in theoretical physics and the result of Gelfand's work on the representation theory of the unitary group and Lie groups in general.

Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and organized a research seminar on the subject.

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    $\begingroup$ But interestingly, he apparently “hotly denied being a mathematical biologist” and spoke of “the unreasonable ineffectiveness of mathematics in the biological sciences” (p. 29). $\endgroup$ – Francois Ziegler Jan 1 at 19:43
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    $\begingroup$ That is interesting! $\endgroup$ – Nik Weaver Jan 1 at 20:43
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What about James Simons, of the Chern–Simons form, and working in topology and with manifolds.

He is perhaps more known for his more applied works, making tons of money from the stock market and funding things related to mathematics (including Quanta Magazine, arxiv, conferences, etc).

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Vladimir Arnold's work ranges from the very abstract (cohomology ring of the colored braid group, symplectic topology, Maslov Index, real algebraic geometry, invariants of plane curves,...) to the applied (stability of the solar system & Arnold diffusion, Cat map, singularity theory,...) and very applied (gömböc, book on Huygens & Barrow, Newton & Hooke,...). Admittedly, his work forms a continuum, but of wide breadth.

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Littlewood is best-known for his pure math research, but during the first World War he worked on ballistics. I suspect there were others who put aside pure math for more applied topics during the wars.

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I have a friend to whom De Rham was famous for completely different reasons — as author of a “corner-cutting algorithm” used in Computer Aided Geometric Design (of car bodies).

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Andrew Gleason solved Hilbert's Fifth Problem, contributed to the foundations of quantum mechanics by proving Gleason's Theorem and was a serious cryptographer during and after WWII.

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Pafnuty Liebovich Chebyshev contributed to probability and number theory, among other matters - but also worked on steam engine "linkage" design. I also remember being told that he had developed a theory of tensile strength of textile strings - but I can't find evidence of that for now.

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I am surprised the name of Bernhard Riemann did not come up already. He founded a few fields of mathematics and it is a bit funny to justify his presence in this list so I'll be short. On the abstract side, his work in number theory. On the applied side, his work on equations of mathematical physics, including hyperbolic equations and propagation of shocks.

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How about Arne Beurling? Complex and harmonic analysis on the "very abstract" side, breaking nazi codes on the "very applied". See https://en.wikipedia.org/wiki/Arne_Beurling and https://bookstore.ams.org/SWCRY/

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James Glimm did his Ph.D. in C $^*$-algebras, where he made long-lasting contributions. He switched fields soon after, and (as Wikipedia says) he has been noted for contributions to C*-algebras, quantum field theory, partial differential equations, fluid dynamics, scientific computing, and the modeling of petroleum reservoirs.

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Thierry Coquand works mainly in formal topology, constructive algebra, and foundations, but he was one of the early authors (and namesake!) of the Coq proof assistant. In fact, most people probably know Coquand only as the creator of Coq but not as a logician/algebraist.

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Eduard Stiefel went from characteristic classes and Lie group representations and topology, to (early) numerical programming and computation of orbits for NASA.

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Claude Elwood Shannon. Do I need to say more?!

 

Shannon has single-handedly both introduced a new mathematical theory, Information Theory and was the author of the first and fundamental results on his Information Theory. Other theories were introduced by a crowd where one or two people stand out and get extra credit. In the case of Shannon's Information Theory, there was nothing like this.

Information Theory has exciting chapters:

  1. data compression;
  2. error-correcting codes;
  3. encryption

Each of these three theories assists the respective technologies.

Shannon is known also for very practical financial activities like making fun of the casino business, where he induced certain casino modifications, etc.

 

Shannon's 1940 PhD Thesis at MIT was entitled An Algebra for Theoretical Genetics, and I've heard that its content kept on getting rediscovered in the 60s by multiple researchers.

 

Kolmogorov had a very high opinion on Shannon. Kolmogorov applied Shannon's ideas (Shannon's entropy function) to solve an old and outstanding problem of classification of Bernoulli shifts (he got half of it, which was a wonderful achievement).

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    $\begingroup$ Answers should be self-contained, so yeah, you should say more... $\endgroup$ – leftaroundabout Jan 2 at 14:14
  • $\begingroup$ But this one is! C'mon, ... $\endgroup$ – Wlod AA Jan 2 at 20:19
  • $\begingroup$ Yeah, ...no. Although you're onto a nice in-joke here, but that would better apply to Kolmogorov rather than Shannon... $\endgroup$ – leftaroundabout Jan 2 at 23:42
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    $\begingroup$ I mostly know Shannon for his paper on communication theory, which certainly had very pure spinoffs in the work of others, but is itself in the "very applied" category. What is the 'pure' work he is known for? $\endgroup$ – Nathaniel Jan 3 at 11:03
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Perhaps Helmut Wielandt might be mentioned. As well as his work on finite group theory, he has a famous theorem on doubly stochastic matrices, and another elegant proof (albeit of a previously known theorem) that the equation $AB -BA = I$ can't hold in any (real or complex) normed algebra.

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Ronald Fisher is considered an establishing figure in the field of modern statistics. A story I've heard goes that his work in statistics was mentioned to a biologist, who responded "He worked in statistics? I knew he was a huge influencer in biology but had no idea he did statistics."

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    $\begingroup$ I believe that his work on design of (agricultural) experiments was a motivation for some of his combinatorial work (on block designs, etc.). $\endgroup$ – Geoff Robinson Jan 1 at 17:55
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    $\begingroup$ On the internet I came across someone who didn't know that James Clerk Maxwell was known for anything besides the invention of color photography. $\endgroup$ – Michael Hardy Jan 2 at 20:22
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    $\begingroup$ With Fisher it can be hard to distinguish between the pure and the applied. Fisher's fiducial methods, introduced in order to solve the Behrens–Fisher problem, is beset with perplexing theoretical questions and with questions in the philosophy of scientific induction, and at the same time the Behrens–Fisher problem is so extremely "applied" that statistics textbooks for people studying biology or psychology or economics, who have no understanding of theoretical mathematics even at the secondary-school level, present an approximate numerical method for dealing with the problem. $\endgroup$ – Michael Hardy Jan 2 at 20:36
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    $\begingroup$ See on rereading Fisher by L Savage. Quote: In addition to Fisher's illustrious career as a statistician he had one almost as illustrious as a population geneticist, so that quite apart from his work in statistics he was a famous, creative, and controversial geneticist. Even today, I occasionally meet geneticists who ask me whether it is true that the great geneticist R. A. Fisher was also an important statistician. Fisher held two chairs in genetics, ( ...) , but was never a professor of statistics. ... $\endgroup$ – kjetil b halvorsen Jan 3 at 17:39
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    $\begingroup$ Another quote from L Savage: Indeed, my recent reading reveals Fisher as much more of a mathematician than I had previously recognized. I had been misled by his own attitude toward mathematicians, especially by his lack of comprehension of, and contempt for, modern abstract tendencies in mathematics (...). Seeing Fisher ignorant of those parts of mathematics in which I was best trained, I long suspected that his mastery of other parts had been exaggerated, but it now seems to me that statistics has never been served by a mathematician stronger in certain directions than Fisher was. $\endgroup$ – kjetil b halvorsen Jan 3 at 17:43

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