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).$\,\!$
 A: 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 Theory of PDE and to fluid mechanics.
A: 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."
A: 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
A: 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. 
A: 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. 
A: 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.
A: 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.
A: Alan Turing is one mathematicians with both “very abstract” and “very applied” achievements: https://en.wikipedia.org/wiki/Alan_Turing
A: 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.
A: Eduard Stiefel went from characteristic classes and Lie group  representations and topology, to (early) numerical programming and computation of orbits for NASA.
A: 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).
A: 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. 
A: 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.
A: 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.
A: 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.
A: 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.
A: 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.
A: 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.
A: 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

A: 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.
A: 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.)
A: 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.
A: 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.


*

*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.

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

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

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

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

A: G. H. Hardy is well-known for his work in number theory, but also for the so called Hardy–Weinberg principle in genomics.
A: Some things Kolmogorov did were fairly "pure", but he is one of the eponyms of the Kolmogorov–Smirnov test.
A: Jerrold Marsden made major contribution to symplectic geometry but also was a key contributor to problems in celestial mechanics and numerical methods.
A: 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.
A: 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.

A: 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.
A: 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. "
A: 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.

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

A: 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.
A: 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.
A: 
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.

A: 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). 
A: 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.
A: 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 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.
A: 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.
A: 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).
A: 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. 
A: 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)

A: 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.
A: 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.
A: 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. 
A: 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/
A: 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.
A: 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." 

A: 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.
A: James Munkres
Very 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 for Linear Assignment problems
A: 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:


*

*data compression;

*error-correcting codes;

*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).
A: George Dantzig is known for his work in Linear Programming, including coining the Simplex algorithm.  But his PhD was accidentally in Statistics.
A: 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.
A: Not sure whether this counts: Martin Hairer has developed a musical soft-ware (Amadeus) which is still used, it seems.
A: 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 achievements in pure mathematics probably don't need to be reviewed here.
A: Alexander Grothendieck
Abstract 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 EGA, SGA, FGA)
Applied achievements: his PhD thesis was in functional analysis, and his early papers focused on "the theory of nuclear spaces as foundational for Schwartz distributions" (among other topics). This work has applications to stochastic PDEs, elliptic PDEs, probability theory and mathematical statistics, physics, engineering, and kernel functions (hence, data analysis and machine learning). Following citations on Google Scholar is a fun way to see citations to Grothendieck's work from a wide variety of application areas. Grothendieck also published a paper (cited 60 times) on solution spaces to a general class of PDEs.
