Mathematicians who made important contributions outside their own field? It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.
I am looking for mathematicians who did just that. An example would be Kurt Gödel's succesful incursion into general relativity.
I am not looking for people who found an application of their work into another field, but for people who, for some reason, had to educate themselves in a domain of which they were previously ignorant, and made some important discovery there.
Edit : I am interested in "modern" examples, let us say "post-Gauss", since it was arguably easier to switch disciplines in ancient times.
 A: Grassmann made his most important contribution in Linguistics.
Poincaré, besides his enormous work in mathematic, also add very important contribution to physics, and his work in philosophy of science was quite influential for a time, and recompensed by a nomination at the French Academy, quite a rare honor (?) for a scientific.
A: Emmanuel Lasker leaps to mind: http://en.wikipedia.org/wiki/Emanuel_Lasker#Academic_activities_1894.E2.80.931918
A: I think that Robert Solovay ought to be mentioned here for his contributions to prime numbers, namely the Solovay–Strassen primality test. 
A: I think big problems in a field being solved by people outside the field in question is often the norm. See for instance the recent solution of the Kadison-Singer conjecture by Marcus-Spielman-Srivastava. The following quote from this AMS Bulletin article of Fulton on Horn's conjecture seems appropriate:

It may be worth pointing out explicitly that although the problems
  solved in this story range over several areas of mathematics —
  including linear algebra, commutative algebra, representation
  theory, intersection theory, and combinatorics — none of the people
  involved in the recent success came to the problems from any of these
  fields. Klyachko came from studying vector bundles on toric varieties;
  Totaro from studying filtered vector spaces using geometric invariant
  theory; Knutson, Agni- hotri and Woodward came from symplectic
  geometry; Tao from harmonic analysis; and Belkale from the study of
  local systems on Riemann surfaces.

A: David Mumford, a well known algebraic geometer, is responsible for the "Mumford-Shah segmentation" model in mathematical imaging. Besides being one of the most cited papers in this area, it also sparked in immense amount of work in geometric measure theory, "special functions of bounded variations", Gamma-convergent numerical approximations and calibration and functional lifting for numerical purposes. To get a glimpse, check the book Singular Sets of Minimizers for the Mumford-Shah Functional.
A: John Tukey's Ph.D. is Mathematics (from Princeton, in Topology).  His contributions are in applied statistics and computation.  Tukey was responsible (in part) for the FFT.
A: I'd mention Ion Barbu, one of the greatest Romanian mathematicians and also one of the greatest Romanian poets. To the trained reader, the structure and feeling of his poems betray his mathematical mindset, a mindset that served in the making of his own lyrical style, which is unique in Romanian literature.
As far as I know, his greatest contribution in the field of mathematics revolves around Barbilian spaces, which bear his name.
Quoting wikipedia:

At the University of Missouri in 1938 L.M. Blumenthal wrote Distance
  Geometry. A Study of the Development of Abstract Metrics, where he
  used the term "Barbilian spaces" for metric spaces based on
  Barbilian's function to obtain their metric.

Wiki for Ion Barbu
A: In his early years, Andrey Kolmogorov was interested in Russian history. Citing Scholarpedia:

He did serious scientific research on XV-XVI century manuscripts concerning agrarian relations in ancient Novgorod. In the twenties he made a hypothesis on the way the upper Pinega was settled, and this conjecture was later confirmed by an expedition to that area.

It is said that he totally moved to mathematics when explained that in history, five different proofs are required for each assertion. There is a related discussion on History Stack Exchange.
A: I think Ronald Fisher's Ph.D. was in mathtematics and he published some things on differential geometry.
He is one of the three major founders of the science of population genetics, with Wright and Haldane.  His writings on biology are fairly voluminous.
He is also the originator of many of the things taught in basic theory-of-statistics courses, including sufficiency and Fisher information.  He single-handedly founded the discipline of design of experiments.
He introduced fiducial inference in order to apply it to what became known as the Behrens–Fisher problem.  It's often hard to tell just what Fisher intended in things he wrote.  Someone named Bartlett in 1936 published a proof that Fisher's fiducial intervals don't have constant coverage rates.  Constant coverage rates are part of the definition of confidence intervals.  Bartlett seemed to suggest that it is therefore an error to use fiducial inference.  Fisher replied that he never intended his intervals to have constant coverage rates, but I haven't been able to figure out just what he did intend.  Bayesian credible intervals also don't have constant coverage rates, but in that case everyone understands why that should be so.  If I knew a bit more than I do about the Behrens–Fisher problem, I'd an write expository paper on why it is not and cannot be really a math problem and any attempt to pretend it is one is a misunderstanding.
A: Probably by "their own field" you mean the whole of mathematics. There are plenty of mathematicians
who work and make important contributions in several fields of mathematics, sometimes very remote from each other.
But here are some outstanding examples of contributions outside of mathematics:

*

*Joseph Fourier (his main contributions to mathematics are "after Gauss", so
I think he qualifies:-)
Actually there are two broad fields where he is considered a "founding father",
Egyptology and medical statistics. Specialists in Egyptology and medical statistics are
even sometimes surprised when I tell them that his contribution to Math was also important:-) But in the city of Grenoble he is mostly remembered as an
outstanding administrator, he was a prefect of the province, and organised many
important improvements in his province.

The examples are really abundant. I am not even mentioning physicists who obtain
Fields medals and mathematicians who obtain Nobel prizes in Economics,
because everyone knows them. Here is a short list of people who come immediately
to my mind.
In XX century:


*Paul Painleve (after the Painleve equations, he switched to aviation and politics,
and even was a prime minister for some time).


*Pierre Fatou. Astronomer by profession. He actually did many observations,
and wrote important papers on celestial mechanics, computation of orbits
and optics of the instruments. His seminal contributions to
pure mathematics are well-known. Probably the last astronomer who was also a great pure mathematician. (This combination was quite common before 19th century).


*Emile Borel (Minister of the Navy in Panleve's government)


*J. von Neumann. (Besides mathematics, computer science, economics, politics).


*Mstislav Keldysh and Mikhail Lavrentjev are considered "the fathers of the Soviet space program".
Keldysh was the "Chief theorist" of the program.
As a pure mathematician
he is famous for deep results in Potential theory and Approximation theory.


*Michail Lavrentjev, one of the famous analysts of the first half of 20-s century,
also contributed a lot to all sorts of applied sciences, and was an important
politician. He created a city in Syberia where all kinds of scientific research is the
main occupation of the inhabitants:-)


*Stanislaw Ulam (a pure mathematician with main interests in set theory, measure theory etc. He also had a patent for the first hydrogen bomb, jointly with E. Teller).


*Israel Gelfand. About 1/2 of his enormous production was in medicine.


*Donald Knuth. Mathematician who created TeX for all of us. You may dispute whether
this is "inside" or "outside" Math:-) As a mathematician he made
an important contribution to combinatorics.
Even among the people I know personally:


*Andrei Gabrielov. If you compare the list of his works on Google Scholar with the list
on Mathscinet, you see that he made almost equally important contributions to
pure Mathematics and Geology.


*Alan Sokal. A famous mathematician and physicist. But most famous for his criticism of
modern philosophers:-)
A: Paul Cohen was an analyst but got a Fields Medal for his work in set theory, proving the independence of the continuum hypothesis from ZFC and the independence of the axiom of choice from ZF.
A: Abraham Robinson was interested in pure mathematics as an undergraduate, but became an expert in aerodynamics during World War II.  After the war, he did his PhD thesis in model theory.
A: If I'm not mistaken, Leonid Kantorovich started as a mathematician then became known as an economist.  (I'm not too familiar with his work; perhaps someone else can edit this answer to give more detail.)
A: Alan Turing, widely regarded as the father of Computer Science, is most widely-cited for his paper The Chemical Basis of Morphogenesis (wikipedia entry). The paper is integral in the development of theoretical biology and chaos theory.
A: In this forum it might be appropriate to mention the contributions of user 766 to signal processing, which led to the discovery of the technique of compressed sensing, and a revolution in medical MRI.
A: Personally I consider John Milnor and Steven Smale as two brilliant mathematicians who contribute in different areas. Milnor contributed to his work in differential topology, K-theory and dynamical systems and Smale contributed to topology and dynamical systems.
A: Thurston (together with Sleator and Tarjan) solved a problem in combinatorics about "rotation" distance between binary trees. The solution translated the problem into a question in hyperbolic geometry, but didn't directly make use of Thurston's previous work in that field. 
A: I think of John Nash. Basically, he is mathematician, specielized in differential geometry, number theory, game theory. And he brought such huge discoveries in economics that he got the Nobel Prize of Economy.
A: Werner Nahm not only made important contributions to conformal field theory, but also conducted research about the Mayan civilization and their astronomy. In his Mayan research, he also worked with Linda Schele and Nikolai Grube and participated in the ongoing decipherment of Maya hieroglyphs.
A: Leonard Adleman (as in RSA Algorithm) made significant contributions to AIDS research.
A: Jean Leray was in differential equations before WWII, but hid his applied mathematics background from his captors at a POW camp, where Leray did some fundamental work on spectral sequences that are widely used in algebraic topology/geometry and other fields today. 
A: Number-theorist G H Hardy is well-known to population biologists for the Hardy-Weinberg Law. 
A: I propose Jim Simons http://en.wikipedia.org/wiki/James_Harris_Simons 
He was a great mathematician, then he became a "big" in economy.
A: You know, a good example in the other direction is John Banzhaf and the Banzhaf index. He studied engineering as an undergraduate and is a practicing attorney, but made a major, albeit anticipated, contribution to Game Theory. (Until checking Wikipedia, I'd never known it was anticipated, and I suspect the original discovery went un-noticed.)
A: James Garfield, in addition to being a published mathematician was also a former President of the United States.  He reformed the civil service system and worked on civil rights issues.
