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Answer by Steven Landsburg for Mathematicians with both “very abstract” and...

Frank Ramsey wrote two papers in economics --- one on optimal taxation and one on optimal savings --- that remain the foundation of both much theoretical work and of much practical policy-making. His...

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Answer by Roland Bacher for Mathematicians with both “very abstract” and...

Not sure whether this counts: Martin Hairer has developed a musical soft-ware (Amadeus) which is still used, it seems.

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Answer by Jim Conant for Mathematicians with both “very abstract” and “very...

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...

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Answer by Xi Li for Mathematicians with both “very abstract” and “very...

The big bird Yuri ManinManin is known for his work in algebraic geometry.He is also father of quantum computing together with Richard Feynman.

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Answer by Kenneth Derus for Mathematicians with both “very abstract” and...

Georg Kreisel did hydrodynamics during and after WWII. Among other things, he determined that the floating harbors used in the D-Day invasion would be stable in heavy seas.

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Answer by Hollis Williams for Mathematicians with both “very abstract” and...

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,...

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Answer by Michael Bächtold for Mathematicians with both “very abstract” and...

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 biologyFunctional labels...

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Answer by Timothy Chow for Mathematicians with both “very abstract” and “very...

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...

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Answer by Francois Ziegler for Mathematicians with both “very abstract” and...

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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Answer by Andreas Blass for Mathematicians with both “very abstract” and...

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)...

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Answer by Francois Ziegler for Mathematicians with both “very abstract” and...

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

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Answer by David Roberts for Mathematicians with both “very abstract” and...

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"...

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Answer by Zhipu 'Wilson' Zhao for Mathematicians with both “very abstract”...

Eugene Dynkin, of probability (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.

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Answer by Dan Fox for Mathematicians with both “very abstract” and “very...

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...

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Answer by Martin Peters for Mathematicians with both “very abstract” and...

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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Answer by Dev Sinha for Mathematicians with both “very abstract” and “very...

Gunnar CarlssonIn 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...

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Answer by David White for Mathematicians with both “very abstract” and “very...

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

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Answer by David White for Mathematicians with both “very abstract” and “very...

Alexander GrothendieckAbstract achievements: his Fields Medal winning work on derived functors, plus a whole new approach to algebraic geometry that has shaped generations of mathematicians after (see...

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Answer by John D. Cook for Mathematicians with both “very abstract” and “very...

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...

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Answer by Gerry Myerson for Mathematicians with both “very abstract” and...

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...

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Answer by slcvtq for Mathematicians with both “very abstract” and “very...

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...

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Answer by Santi Spadaro for Mathematicians with both “very abstract” and...

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...

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Answer by Goldstern for Mathematicians with both “very abstract” and “very...

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...

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Answer by John Coleman for Mathematicians with both “very abstract” and “very...

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...

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Answer by Wlod AA for Mathematicians with both “very abstract” and “very...

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.

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Answer by Piero D'Ancona for Mathematicians with both “very abstract” and...

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...

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Answer by Mars for Mathematicians with both “very abstract” and “very...

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...

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Answer by relean elo for Mathematicians with both “very abstract” and “very...

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...

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Answer by Francois Ziegler for Mathematicians with both “very abstract” and...

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...

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Answer by Yuval Peres for Mathematicians with both “very abstract” and “very...

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

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Answer by Wlod AA for Mathematicians with both “very abstract” and “very...

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...

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Answer by user6976 for Mathematicians with both “very abstract” and “very...

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...

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Answer by Wlod AA for Mathematicians with both “very abstract” and “very...

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,...

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Answer by Wlod AA for Mathematicians with both “very abstract” and “very...

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...

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Answer by Wlod AA for Mathematicians with both “very abstract” and “very...

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...

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Answer by David Eppstein for Mathematicians with both “very abstract” and...

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...

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Answer by Ethan Bolker for Mathematicians with both “very abstract” and “very...

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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Answer by Igor Rivin for Mathematicians with both “very abstract” and “very...

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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Answer by Francois Ziegler for Mathematicians with both “very abstract” and...

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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Answer by Nik Weaver for Mathematicians with both “very abstract” and “very...

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...

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Answer by Per Alexandersson for Mathematicians with both “very abstract” and...

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...

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Answer by Nik Weaver for Mathematicians with both “very abstract” and “very...

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...

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Answer by Mark L. Stone for Mathematicians with both “very abstract” and...

James MunkresVery abstract: Obstructions to the smoothing of piecewise-differentiable homeomorphisms (doi:10.1090/S0002-9904-1959-10345-1)Very applied: Munkres assignment (a.k.a. Hungarian) algorithm...

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Answer by Piyush Grover for Mathematicians with both “very abstract” and...

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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Answer by Thomas Sauvaget for Mathematicians with both “very abstract” and...

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...

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Answer by Francois Ziegler for Mathematicians with both “very abstract” and...

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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Answer by polfosol for Mathematicians with both “very abstract” and “very...

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...

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Answer by user131781 for Mathematicians with both “very abstract” and “very...

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...

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Answer by bof for Mathematicians with both “very abstract” and “very applied”...

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...

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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...

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