# Mathematicians with both “very abstract” and “very applied” achievements

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).$$\,\!$$

• 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. – Wlod AA Jan 2 at 0:27
• I asked exactly the same question elsewhere and got shut down with almost no answers. mathoverflow.net/questions/345636/… – Hollis Williams Jan 2 at 19:35
• @Tom, that' MO-life/game, it comes with the territory. – Wlod AA Jan 2 at 20:17
• @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. – Tom Copeland Jan 2 at 23:57
• 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.) – Joel Reyes Noche Jan 3 at 2:12

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.

Noga Alon has hundreds of contributions in combinatorics, but also co-authored the foundational paper on streaming algorithms that has been cited more than 1800 times according to Google Scholar:

Alon, Noga, Yossi Matias, and Mario Szegedy. "The space complexity of approximating the frequency moments." Journal of Computer and system sciences 58, no. 1 (1999): 137-147.

Sam Karlin and Rick Durrett are among the leading probabilists who also contributed to Mathematical biology.

Olga_Ladyzhenskaya made fundamental contributions to the Thepry of PDE and to fluid mechanics.

• that foundational paper looks very theoretical – mathworker21 Jan 2 at 19:43

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.

• 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',... – Lennart Meier Jan 3 at 20:42

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.

• This is the same Solovay as the Solovay-Strassen primality test, is it not? – Erick Wong Jan 7 at 3:03
• @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). – Andreas Blass Jan 19 at 15:29
• 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!" – Andreas Blass Jan 19 at 15:35
• @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?). – Goldstern Jan 25 at 11:59

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 ... $$\to \infty$$.

Gunnar Carlsson

In pure math, he works in homotopy theory, having resolved the Segal Conjecture, as well as in manifold topology, with cases of Borel and Novikov conjectures, and also in algebraic K-theory.

In applied math, he is one of the founders of the field of topological data analysis and among the first (if not the first) to develop persistent homology. With Gurjeet Singh he created the Mapper algorithm based on the Reeb graph. That has been used in many, many settings, for example finding a new genetic marker for breast cancer. He co-founded the company Ayasdi, which continues to develop topological (and topology-inspired) tools along often in conjunction with machine learning, with clientele which ranges from academics to government to industry in an impressive array of areas.

Shing-Tung Yau, well-known to pure mathematicians for the proof of the Calabi conjecture, the Donaldson-Uhlenbeck-Yau theorem, the positive mass theorem in general relativity, and differential Harnack inequalities and various gradient estimates for PDE, also has interesting papers in applied topics such as

• "Genus zero surface conformal mapping and its application to brain surface mapping" (with X. Gu, Y. Wang, T.F. Chan, and P.M. Thompson)
• "GPU-assisted high resolution, real-time 3-D shape measurement" (with S. Zhang and D. Royer)
• "Geometric understanding of deep learning" (with N. Lei, Z. Luo, and X. Gu)
• I am not sure why this answer has been downvoted. Yau is an author of many more papers on imaging, discrete Ricci flow, graph theory, etc. than those listed, and some of these papers are more legitimately applied works than some of the contributions of other mathematicians mentioned here and upvoted. To put a randomly chosen example: how is sciencedirect.com/science/article/pii/S0167839618301249 not an applied paper? – Dan Fox Jan 8 at 17:04

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.

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.

• Zermelo also started as assistant to Max Planck, publishing papers on thermodynamics in a famous polemic with L. Boltzmann 1895–1896. – Francois Ziegler May 4 at 23:23
• @FrancoisZiegler Indeed, and this is easily accessible in his Collected Works. See also Heinz-Dieter Ebbinghaus´ biography of Zermelo. – Martin Peters May 6 at 5:58

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.

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.

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.

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).

• 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. – Chan Bae Jan 2 at 6:06
• Indeed there is a recent and interesting book on the subject (József Illy, 2012). – Francois Ziegler Jan 3 at 15:51

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.

• 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 – slcvtq Jan 3 at 14:24

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.

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.

Just came across a page of 25+ of Mark Goresky’s Engineering publications.

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.

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.

• 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 – Henry Jan 1 at 22:15

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

• I think you mean symplectic geometry — which originally was celestial mechanics. – Francois Ziegler Jan 1 at 18:11

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.

While his other family members outdid Andrzej Trybulec in topology and geometry, he added to these specializations also his creation of Mizar -- the computer proof-checker.

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.

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.

Abstract achievements: his Fields Medal winning work on algebraic $$K$$-theory, plus inventing model categories, homotopical algebra, and an axiomatic approach to abstract homotopy theory.

Applied achievements: his 1964 PhD thesis was about partial differential equations, and was used in many subsequent applied papers on systems of PDEs, including an Annals paper by Goldschmidt. Later, in the 1980s, he did work in Riemannian geometry and functional analysis and he invented the notion of 'superconnection' in differential geometry and analysis. This work has been applied to work on Heat kernels and Dirac operators, Deformation quantization, elliptic operators, index theory, and topological quantum field theory.

• Just as with Grothendieck - just because his work had been USED by applied people, does not make it applied. – Igor Rivin Jan 6 at 21:34
• I respectfully disagree. Both Grothendieck and Quillen published highly cited papers in applied mathematics. Just because I emphasized further applications by later authors does not mean their papers were not already applied. I leave it to experts to decide (via their votes) if these contributions were "very applied" or "pretty abstract" as you wrote on the Grothendieck answer. – David White Jan 6 at 22:18
• @DavidWhite, I have to agree with Igor. Both Quillen and Goldschmidt’s work on overdetermined systems of PDE are highly abstract and are quite distant from applied math. All of the citations are from equally abstract pure math papers. I do not know of any applied math papers citing their work. It is in fact difficult to find any nontrivial interesting examples, other than already well known ones, of the type of systems they study. – Deane Yang Jan 14 at 7:10
• Thanks for this additional context. It was fun for me, as a pure mathematician, to learn a bit about the applied contributions of Quillen and Grothendieck. I'm not surprised that those contributions are of a more theoretical nature, and happy to hear it from experts. – David White Jan 14 at 14:15

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.

• 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). – Francois Ziegler Jan 2 at 6:03
• It would hardly be spam it is such an interesting question! Better that someone with ample reputation ask it, too. – R Hahn Jan 2 at 6:14
• I’ll let someone else do it, if they feel the subquestion is worth splitting off. – Francois Ziegler Jan 2 at 6:19
• Quite a bit of mathematical work/papers (or non-mathematical work by mathematicians) done for DOD contractors were never classified. – Wlod AA Jan 2 at 10:14
• As I note in another answer, Hassler Whitney also worked on fire control during the war. – Gerry Myerson Jan 5 at 4:02

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.

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

George Dantzig is known for his work in Linear Programming, including coining the Simplex algorithm. But his PhD was accidentally in Statistics.

• Both of which are applied by any means, no? – Dirk Jan 1 at 14:16

Eugene Dynkin, of prpbability (Dynkin’s Lemma, among many other things) and Lie algebra (Dynkin Diagrams, in fact according to Wikipedia the whole positive root formalism is worked or by him) fame.