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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
    Commented Jan 2, 2020 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
    Commented Jan 2, 2020 at 19:35
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    $\begingroup$ @Tom, that' MO-life/game, it comes with the territory. $\endgroup$
    – Wlod AA
    Commented Jan 2, 2020 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
    Commented Jan 2, 2020 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.) $\endgroup$
    – JRN
    Commented Jan 3, 2020 at 2:12

68 Answers 68

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Another example is Piotr Novikov. He started as a set theorist, then logician and group theorist (where he is famous for the work on the word problem for groups, and the Burnside problem). But he is also well known for the solution of the inverse potential problem for star-shaped objects. This is used for finding iron ore deposits. I think that Novikov's theorem is still the most general and widely used result in the inverse potential theory.

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I am not sure what "Robert Solovay's checksum utility" is, but it sounds very applied, and is mentioned in hundreds of LaTeX input files.

He is also one of the giants of set theory, perhaps best known for his model of set-theory in which every set of reals is Lebesgue measurable.

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    $\begingroup$ This is the same Solovay as the Solovay-Strassen primality test, is it not? $\endgroup$
    – Erick Wong
    Commented Jan 7, 2020 at 3:03
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    $\begingroup$ @ErickWong Yes. And also the same as Solovay as the Solovay-Kitaev theorem. That theorem describes how uniformly a finite;y generated subgroup fills up $SU(2)$. That sounds very pure, but it has major implications in quantum computing (taking the subgroup's generators to be quantum operations for which good error-correction is available). $\endgroup$
    – Andreas Blass
    Commented Jan 19, 2020 at 15:29
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    $\begingroup$ Although the Lebesgue-measure model is surely Solovay's best-known set-theoretic achievement, I (and I suspect also some other set theorists) consider his and Dana Scott's Boolean-valued models at least equally important. At a first-year grad student in spring of 1967, I audited a class by Tony Martin on independence results. Most of the course was based on Cohen's book, but at the very end Tony briefly described Boolean-valued models. I immediately thought "Oh, so that's what this semester has really been about!" $\endgroup$
    – Andreas Blass
    Commented Jan 19, 2020 at 15:35
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    $\begingroup$ @AndreasBlass Agreed. It did not occur to me to mention Boolean-valued models as his most important result because (at least for me, and perhaps for my generation) they look so natural (dare I say obvious?). $\endgroup$
    – Goldstern
    Commented Jan 25, 2020 at 11:59
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Norbert Wiener -- well known for his profound mathematics but also as the father of cybernetics.

"Wiener is considered the originator of cybernetics, a formalization of the notion of feedback, with implications for engineering, systems control, computer science, biology, neuroscience, philosophy, and the organization of society.

Norbert Wiener is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could possibly be simulated by machines and was an important early step towards the development of modern artificial intelligence."

Also:

"During World War II, his work on the automatic aiming and firing of anti-aircraft guns caused Wiener to investigate information theory independently of Claude Shannon and to invent the Wiener filter. (To him is due the now standard practice of modeling an information source as a random process—in other words, as a variety of noise.) His anti-aircraft work eventually led him to formulate cybernetics."

Also ... $\to \infty$.

See: https://en.wikipedia.org/wiki/Norbert_Wiener

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  • $\begingroup$ Wiener’s Cybernetics sounds rather abstract to me...? $\endgroup$
    – Francois Ziegler
    Commented Jan 3, 2020 at 14:25
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Raoul Bott. The Bott-Duffin theorem, which is essentially the result of Bott's doctoral thesis (in electrical engineering; the director was Richard Duffin), gives a constructive proof that a positive-real function is the impedance of a transformerless network. This is a basic result in electrical engineering and control theory. Bott's many accomplishments as a topologist are well known.

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Surprisingly to me, Garrett Birkhoff also did some very applied mathematics (Wikipedia says "During and after World War II, Birkhoff's interests gravitated towards what he called "engineering" mathematics."). I have his book Hydrodynamics in front on me, and it has plots of experimentally-determined results, photos of objects plunging through water, but also sections on group theory and a Lie algebras. Of course, he also coauthored Algebra with Mac Lane, and is well-known for lattice-theoretic work. He also worked on computational mathematics and numerical linear algebra.

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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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  • $\begingroup$ I think that they are quite the same thing. The theory behind Coq is quite foundational and logic. $\endgroup$
    – Z. M
    Commented Jul 9, 2022 at 19:51
  • $\begingroup$ @Z.M The theory, sure; a lot of type theory work was driven by Coq. But according to my understanding, Coquand was also involved in a lot of the "dirty" implementation work. $\endgroup$
    – xuq01
    Commented Jul 10, 2022 at 4:05
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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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    $\begingroup$ Apologies, this duplicates an answer at @Tom’s question which I hadn’t noticed. $\endgroup$
    – Francois Ziegler
    Commented Jan 3, 2020 at 14:11
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Albert Einstein, in addition, to be a colossus in Physics, had patents (inventions), and he had a significant contribution to Differential Geometry (and, on the top of it, also to tensor analysis, including Einstein notation).

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    $\begingroup$ One of his patents was for a refrigerator design! His "very abstract" may have been less abstract than a lot of the people mentioned in other answers, but his "very applied" was a lot more applied than, say, compressed sensing. $\endgroup$
    – Solveit
    Commented Jan 2, 2020 at 6:06
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    $\begingroup$ Indeed there is a recent and interesting book on the subject (József Illy, 2012). $\endgroup$
    – Francois Ziegler
    Commented Jan 3, 2020 at 15:51
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There is Ernst Zermelo, who is well-known for his work in logic, but who was also a pioneer in optimisation and what is now called control theory.

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    $\begingroup$ Zermelo also started as assistant to Max Planck, publishing papers on thermodynamics in a famous polemic with L. Boltzmann 1895–1896. $\endgroup$
    – Francois Ziegler
    Commented May 4, 2020 at 23:23
  • $\begingroup$ @FrancoisZiegler Indeed, and this is easily accessible in his Collected Works. See also Heinz-Dieter Ebbinghaus´ biography of Zermelo. $\endgroup$
    – Martin Peters
    Commented May 6, 2020 at 5:58
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I am slightly surprised that Henri Poincaré is not already on this list. Perhaps it is because almost all of his work could be considered applied mathematics. But his contributions to the foundations of algebraic topology were extremely important, and seem "pure" to me.

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  • $\begingroup$ Poincaré is also the pioneer of automorphic forms. I don't know whether this is pure. $\endgroup$
    – Z. M
    Commented Jul 9, 2022 at 19:55
  • $\begingroup$ Poincar\'e was also a mining engineer. $\endgroup$
    – Roland Bacher
    Commented Aug 5, 2022 at 8:59
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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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Michel Demazure worked on group schemes as a member of Bourbaki. But he is also known for his work in computer vision for recovering the 3d geometry of a scene by comparing the positions of known points in two still photos of the scene.

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Misha Gromov has written on the formalization of genetic and biomolecular structures and the thinking process. Some articles from his website:

  • Mathematical slices of molecular biology
  • Functional labels and syntactic entropy on DNA strings and proteins
  • Pattern formation in Biology, Vision and Dynamics
  • Crystals, proteins, stability and isoperimetry
  • Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1,2
  • Structures, Learning and Ergosystems Chapters 1-4, 6
  • Memorandum Ergo
  • Math Currents in the Brain
  • Learning & Understanding, chapter 1,2
  • Great Circle of Mysteries: Mathematics, the World, the Mind
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Dana Scott's achievements include work in pure set theory and also work in computer science. He proved that there are no measurable cardinals in Gödel's constructible universe and (with Solovay) developed the Boolean-valued-model view of forcing. He also introduced Scott domains (though not with that name) for denotational semantics of programming languages.

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John McCleary will give a talk at the JMM in a couple of weeks on "Hassler Whitney and Fire Control in WWII." Whitney "was assigned to work on fire control, the mathematics of aiming weapons for accuracy." Here's the abstract.

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Arguably, the greatest ever mathematical logician, Emil Leon Post, was among the main founders of Computer Science (Informatics/Informatique).

The fate was cruel to him, in more than one way, hence no wonder that he is vastly unappreciated.

Emil Post had significant contributions to algebra too. But let's concentrate on mathematical logic.

  1. People don't appreciate the Emil Post's theorem about the elementary logic: tautologies = theorems. It may seem trivial but there are hardly any textbooks which include a complete(!!) proof. There is an objective reason why this theorem is not trivial. Indeed, a minor modification of the axioms of Boolean algebra leads to systems which are very hard to tell from actual Boolean algebras. On occasions, it takes intensive computer programs to decide the issue.

  2. Emil Post had developed formalization independently of David Hilbert (there are trade-offs between the approaches by these two mathematicians).

  3. Emil Post has proved the incompleteness theorem years before Kurt Gödel (again, there were trade-offs between the two).

  4. Emil Post has developed the theory of algorithms independently of Alan Turing; occasionally, people talk about Post-Turing machines.

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  • $\begingroup$ What work are you considering applied? $\endgroup$
    – usul
    Commented Apr 23, 2022 at 4:10
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As I have been reading Rota's Indiscrete thoughts lately I had in mind the following mathematician, Jacob T. Schwartz.

Citing from the book,

If a twentieth century version of Emerson's Representative Men were ever to be written, Jack Schwartz would be the subject of one of the chapters. The achievements in the exact sciences of the period that runs from roughly 1930 to 1990 may well remain unmatched in any foreseeable future. Jack Schwartz' name will be remembered as a beacon of this age. No one among the living has left as broad and deep a mark in as many areas of pure and applied mathematics, in computer science, economics, physics, as well as in fields which ignorance prevents me from naming.

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    $\begingroup$ Let me mention that this is the Schwartz from Dunford--Schwartz. Some widely cited articles in other fields include 'On the Existence and Synthesis of Multifinger Positive Grips' (Robotics), 'Probabilistic algorithms for verification of polynomial identities', 'Ultracomputers', 'Affine invariant model-based object recognition', 'On meaning', 'Programming with sets: An introduction to SETL',... $\endgroup$
    – Lennart Meier
    Commented Jan 3, 2020 at 20:42
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Ronald Graham spent his career at Bell Labs working on applied problems such as scheduling theory, but is also known for his work in Ramsey theory. In that context Graham's number held the record for many years as the largest natural number ever used in a serious mathematical proof. Such numbers are not really part of applied mathematics.

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Stephen Smale, who is mainly known for his contributions to topology and topological dynamics, also did important work in mathematical economics.

Smale, Steve, Global analysis and economics, Handbook of mathematical economics, Vol. 1, 331-370 (1981). ZBL0477.90014.

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    $\begingroup$ It seems like the field of economics that Smale contributed to, general equilibrium theory, is just as non-applied in its nature as his work in topology. His more recent work in biology and learning might be more relevant: "Emergent behavior in flocks" and "On the mathematical foundations of learning" both by Cucker and Smale, and "Learning theory estimates via integral operators and their approximations" and "Estimating the approximation error in learning theorry" and "Shannon sampling and function reconstruction from point values" all by Ding-Xuan Zhou and Smale $\endgroup$
    – slcvtq
    Commented Jan 3, 2020 at 14:24
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Just came across a page of 25+ of Mark Goresky’s Engineering publications.

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Mikhail L. Zeitlin, or Gel’fand-Zeitlin basis fame (1950), later switched to “game theory, the theory of automata, computer science, physiology, and mathematical methods of biology”.

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The big bird Yuri Manin

Manin is known for his work in algebraic geometry.

He is also father of quantum computing together with Richard Feynman.

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Richard Arenstorf worked in number theory and in orbital mechanics. He is best remembered for the Arenstorf orbit used by the Apollo program.

His pure and applied work weren’t that far apart. Both involved a lot of classical analysis. He struck me as sort of a 19th century mathematician, even though he was born in 1929.

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Frank Garside (he doesn't have a wikipedia page but he has this https://en.wikipedia.org/wiki/Garside_element) was responsible for solving the conjugacy problem in the braid group, and then became the mayor of Oxford.

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    $\begingroup$ George Reid was Senior Wrangler and then an algebraist before becoming Mayor of Cambridge in 1990-91. I am not sure this type of example is what the question was looking for $\endgroup$
    – Henry
    Commented Jan 1, 2020 at 22:15
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Richard von Mises did seminal work on the philosophical foundations of probability in terms of long-run frequencies starting in the 1930s. This led to a series of attempts to revise his approach to fix problems, culminating in concepts of algorithmic randomness based on computability from Chaitin, Kolmogorov, Martin-Löf, etc.

Von Mises was also an engineer who is known for contributions to aerodynamics and solid mechanics.

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Jerrold Marsden made major contribution to symplectic geometry but also was a key contributor to problems in celestial mechanics and numerical methods.

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    $\begingroup$ I think you mean symplectic geometry — which originally was celestial mechanics. $\endgroup$
    – Francois Ziegler
    Commented Jan 1, 2020 at 18:11
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The person with the most citations with "mathematics" on Google Scholar is Eric Lander. He started as a representation theorist, and then moved into molecular biology and genetics.

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Several answers (on Beurling, Gleason, Gröbner, Littlewood, Rankin, Robinson, Turing, Ulam, Whitney) suggest that applied work was often classified. I also heard about Vieta being his King’s cryptographer and Monge’s first work being classified.

Two quotes to illustrate that sometimes this means “interesting achievement”:

(Notices AMS 63, p. 505): Above are excerpts from two Nash letters that the National Security Agency (NSA) declassified and made public in 2012. In these extraordinary letters sent to the agency in 1955, Nash anticipated ideas that now pervade modern cryptography and that led to the new field of complexity theory. (In the obituary for Nash that appears in this issue of the Notices, page 492, John Milnor devotes a paragraph to these letters.)

and sometimes apparently not:  

(Mac Lane 1976, p. 138): faute de mieux, finds himself in New York as Director of the Applied Mathematics Group of Columbia University, instructed to hire many fresh mathematical brains to help with the research side of the war effort. One of his first acts was to hire Samuel Eilenberg—as well as Irving Kaplansky, George Mackey, Donald Ling, and many others. During the day we all worked hard at airborne fire control... [The] report (more exactly, part 2 on “Aerial Gunnery Problems,” as cited in the bibliography) was initially classified confidential and hence buried in the Government Archives. By now it is declassified, but hardly interesting.

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    $\begingroup$ I’d be curious about more examples of declassified work that turned out significant (and would happily move this answer there if someone asks — I just hesitate to spam the site with another big-list question myself). $\endgroup$
    – Francois Ziegler
    Commented Jan 2, 2020 at 6:03
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    $\begingroup$ It would hardly be spam it is such an interesting question! Better that someone with ample reputation ask it, too. $\endgroup$
    – R Hahn
    Commented Jan 2, 2020 at 6:14
  • $\begingroup$ I’ll let someone else do it, if they feel the subquestion is worth splitting off. $\endgroup$
    – Francois Ziegler
    Commented Jan 2, 2020 at 6:19
  • $\begingroup$ Quite a bit of mathematical work/papers (or non-mathematical work by mathematicians) done for DOD contractors were never classified. $\endgroup$
    – Wlod AA
    Commented Jan 2, 2020 at 10:14
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    $\begingroup$ As I note in another answer, Hassler Whitney also worked on fire control during the war. $\endgroup$
    – Gerry Myerson
    Commented Jan 5, 2020 at 4:02
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A similar story to that of Leray is David Gilbarg. He originally did his PhD on algebraic number theory with Emil Artin, but then switched to more applied topics because of the Second World War, becoming more well-known for work on PDE theory and fluid dynamics.

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William Tutte. He is well known for his contributions to graph and matroid theory, including pioneering the enumeration of planar graphs, and introducing the so called Tutte polynomial. He is less well known for his work on deciphering German codes during World War II, similar to Turing. According to Wikipedia "During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. "

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