Notable mathematics during World War II It seems fairly well known that Leray originated the ideas of spectral sequences and sheaves while being held in a prisoner of war camp in Austria from 1940 to 1945.  Weil famously proved the Riemann hypothesis for curves in 1940, while in prison for failure to report for army duty.  I recently learned that Linnik's famous theorem on primes in arithmetic progressions was published in 1944, just after the siege of Leningrad ended.  So now I would like to ask:

What are some other examples of notable mathematics done during World War II?

 A: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002281287
Gentzen published this paper in 1943 which initiated ordinal proof theory. I find it quite remarkable that he (Gentzen) could continue his logical studies after 1933, although Bieberbach obsessively tried to establish his 'German mathematics', a strange product of racism and misinterpreted intuitionism.
A: Onsager's solution of the 2-dimensional Ising model of ferromagnetism:
https://en.wikipedia.org/wiki/Ising_model
A: To complete Tolland's answer, John von Neumann was the leading mathematician in Manhattan project. In this context, he started the mathematical analysis of multi-dimensional shock waves in the Euler equations of gas dynamics.
A: Don't forget the cryptography work done by Turing, Welchman, and others during the war.  The "Theorem that won World War II" (Rejewski's original group-theoretic attack on the Enigma encryption) was actually done shortly before the war, though. 
A: During the Second World War the theory of stochastic observation of a time-invariant process was developed by Wiener in the US and Kolmogorov in the USSR almost simultaneously. The results were published in a classified report which was declassified after the war, "Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications".
A: Kolmogorov in 1941 found his famous 5/3 law for the energy distribution in the turbulent fluid. It was one of the few exact results on turbulent flow in his time.
A: If theoretical physics counts, Sin-Itiro Tomonaga worked out his version of quantum electrodynamics in Japan during the war.  He shared the 1965 physics Nobel with Feynman and Schwinger for it.
A: Grothendieck went to Vietnam to deliver lectures and a report of what he did can still be found online.
Bertrand Russell was imprisoned during WWI for anti-war activities and wrote "Introduction to Mathematical Philosophy" (1919) while in prison.
Hardy, in protest at Russell's consequent dismissal from Cambridge, left Cambridge to Oxford and continued working there and collaborating by mail with Littlewood. Both of them worked during that time in Mathematics and there is a work of fiction written about it.
A: Since Laurent Schwartz  received his Fields Medal in 1950 for his work on distributions, it is reasonable to assume that the bulk was done during WW II. This is confirmed by Treves' obituary,
A: Operations research was developed under WWII! This is mentioned in other answers, but only as "mathematical programming", while OR is much wider than that.  One paper says 
"  Operations Research is a ‘war baby’. It is because, the first problem attempted to solve in a
systematic way was concerned with how to set the time fuse bomb to be dropped from an aircraft on
to a submarine. In fact the main origin of Operations Research was during the Second World War. "
googling for "operations research second world war" (or throw into that "submarine") gives a lot of information, one example which looks interesting is
http://www.ibiblio.org/hyperwar/USN/rep/ASW-51/index.html
which is an statistical analysis of anti-submarine warfare.
A: On the other side of the war, Teichmüller did some of his best work during World War II.  According to the MacTutor biography, he volunteered to serve on the Eastern Front in 1943 and got killed.  My impression, then, is that his Nazi fanaticism was a crime against his own mathematical career as well as against other mathematicians.
A: Not World War II, but World War I:
The 1st Edition of Abraham Fraenkel’s book Einleitung in die Mengenlehre (Introduction to Set Theory) went to press during World War I. Fraenkel had been teaching set theory to his comrades while at war, and this book was his lecture notes, so to say. He also gave his venia legendi lecture during the war, while on furlough.
A: The statistician John Kerrich conducted various probability experiments while in a German internment camp in Denmark during World War Two. Most famously, he flipped a coin 10,000 times and analyzed the results. Perhaps not "notable mathematics" in the way the question intended, but a notable achievement all the same. If nothing else, it makes a nice story for a low-level stats class.
A: While Claude Shannon was working on cryptography during the war, he worked out the key principles of information theory.  Wikipedia (https://en.wikipedia.org/wiki/Claude_Shannon#Wartime_research) explains how these ideas were gradually published after the war:
At the close of the war, he prepared a classified memorandum for Bell Telephone Labs entitled "A Mathematical Theory of Cryptography", dated September 1945. A declassified version of this paper was published in 1949 as "Communication Theory of Secrecy Systems" in the Bell System Technical Journal. This paper incorporated many of the concepts and mathematical formulations that also appeared in his "A Mathematical Theory of Communication" [1948]. Shannon said that his wartime insights into communication theory and cryptography developed simultaneously and that "they were so close together you couldn’t separate them".[20] In a footnote near the beginning of the classified report, Shannon announced his intention to "develop these results … in a forthcoming memorandum on the transmission of information."[21]
A: In the opposite direction, here is an example (one of presumably hundreds of others) of work that got cut short by the war.
The dedication before the introduction reads:

Flight-Lieutenant P. R. Taylor was missing, believed killed, on active service in November 1943. The editors express their thanks to Mr. J. E. Rees for arranging the paper from the original manuscript and to Professor Titchmarsh for revising and completing the argument.

A: The story of Wolfgang Doeblin. Results remained unknown till 2000.
See "Comments on the life and mathematical legacy of Wolfgang Doeblin",
by Bernard Bru and Marc Yor  (link)
There is also a documentary.
A: Eilenberg and Mac Lane's papers on category theory started appearing: "Natural Isomorphisms in Group Theory" in the Proc. National Acad. Sci. USA in 1942 and "General Theory of Natural Equivalences" in Transactions of the AMS in 1945. 
That doesn't quite fit David's request for work done in wartime conditions.  Mathematicians in the US were not exactly under siege! A more suitable example would be the Gelfand--Naimark theorem characterizing C*-algebras and the Gelfand--Raikov theorem showing that the points in any locally compact group can be separated by some irreducible unitary representation of the group.  These both appeared in 1943. 
A: George Dantzig essentially developed the foundations of linear programming while he was under the employment of the military. As has been mentioned in books, the term "programming" itself in this context is military terminology. (The simplex method however came after the war, in 1947).
A: Zariski started using abstract algebra to develop algebraic geometry in the late 1930's, and a lot of his major work was done during the war itself, such as his papers on resolution of singularities. 
A: Supposedly, after the war had ended, Siegel asked Harald Bohr what had happened in mathematics in Europe during the war. Bohr responded: "Selberg."
Google: "Siegel Bohr Selberg" and you can find a number of references to the quote.
A: I remember reading a interesting article from the AMS a while ago about the Japanese mathematician Mikio Sato, who independently did some important work in algebraic analysis during the World War II. If my memory serves me well he was developing his theory of hyperfunctions at a young age all the while having to feed and protect his family during the war and "carrying coal" to earn a living. Here is a link to the AMS article: http://www.ams.org/notices/200702/fea-sato-2.pdf
Edit: Since it hasn't yet been mentioned, Alan Turing did great work during WW-II: he participated in a team that cracked the Enigma machine and many other codes/cyphers. https://en.wikipedia.org/wiki/Alan_Turing
A: Monte Carlo integration was first put to use during the Manhattan project.
A: The paper
M. L. Cartwright, J. E. Littlewood. On non-linear differential equations of the second
order. I. The equation $y''-k(1-y^2)y+y=b\lambda k\cos(\lambda t+a)$ J.London Math.
Soc. 20, (1945)
was not only written during the war, but also was stimulated by the war. Subsequently it played an important role in prehistory of hyperbolic dynamics.
In 1960 Stephen Smale conjectured that Morse-Smale systems are the only structurally
stable systems.
It was pointed out to Smale that his conjectures are likely to be false. Rene
Thom argued that hyperbolic automorphism does not lie in the closure of Morse-
Smale systems. Norman Levinson wrote to Smale with a reference to the above paper in
which Cartwright and Littlewood studied certain differential equation of second
order with periodic forcing. This work arose from war-related studies involving
radio waves. The equation leads to a flow on R3. According to Levinson this 
flow
has infinitely many periodic orbits; this phenomenon is robust which can be seen
from the paper and also it was directly proved for a dierent equation in his own
work. This led Smale to discovery of the famous horseshoe and subsequent explosive development in smooth dynamics.
A: Hochschild was working at Aberdeen Proving Ground in 1944 when he wrote "On the cohomology groups of an associative algebra" which was published in the Annals in '45.
A: Of course Switzerland was one of the few countries where mathematicians could basically do their business as usual, during WW2. Many fundamental discoveries of the Zurich school on algebraic topology (Hopf, Stiefel, Eckmann...) took place during this period. The journal Commentarii Mathematici Helvetici was published without interruption, and it is worth having a look at the Tables of contents (see e.g. http://retro.seals.ch/digbib/en/vollist?UID=comahe-001,comahe-002,comahe-003) to see that it was probably the best european journal during the wartime period.
A: Someone had told me that the person who invented the "Stalk" of a Sheaf coined the term inside a concentration camp.
I can't confirm this though so please let me know if I am right.
https://en.wikipedia.org/wiki/Sheaf_(mathematics)
A: In 1944 Kiyosi Itô published his seminal paper 'Stochastic integral' in Proc. Imp. Acad. Tokyo, MathSciNet link, doi:10.3792/pia/1195572786. Shortly he initiated the study of stochastic integral equation in 1946 and obtained the noted lemma in 1951. See his MacTutor Biography and more about the history of stochastic calculus in Jarrow and Protter (2004).
A: The McCulloch Pitts paper "A Logical Calculus of the Ideas Immanent in Nervous Activity" came out in 1943.
https://web.csulb.edu/~cwallis/382/readings/482/mccolloch.logical.calculus.ideas.1943.pdf
A: In a series of papers published during and just after World War II, Richard Brauer hit his stride in pioneering the modular character theory of finite groups, and its relation to ordinary (complex) character theory and the structure of finite groups. Highlights include the "Main Theorems" of block theory, and the analysis of groups whose order is divisible by some prime to the first power.
A: Jean Leray did much of his notable work, such as introducing sheaves and spectral sequences, while in a prisoner of war camp during World War II.
A: Alan Turing is an obvious answer
A: Albert Gloden's book, Mehrgradige Gleichungen, was published in Groningen in 1944. Some of it is out of date, but it's still a good place to start the study of multigrade equations (equations in integers $$a_1^r+a_2^r+\cdots+a_n^r=b_1^r+b_2^r+\cdots+b_n^r$$ that hold for several values of $r$).
A: Abraham Wald’s work on Sequential Analysis was largely done during World War II, and largely motivated by it. (There were definitely applications of the work during the war, even if some of the proofs were published afterwards.)  Bill Casselman’s article (http://www.ams.org/publicoutreach/feature-column/fc-2016-06) is one source on this, with a link to several other sources on the topic.
A: Ernst Witt claimed that he had discovered the Leech lattice in 1940, see e.g.
https://en.wikipedia.org/wiki/Leech_lattice
A: Notable for the circumstances, from the book The Universal Computer by Martin Davis:
A biography of Leibniz "was completed by Professor Kurt Huber in prison while awaiting execution by the Nazis. He had supported the efforts of his students at the University of Munich who had formed the "White Rose" underground group and who were decapitated for distributing anti-Nazi leaflets."
Huber, K. Leibniz. Munich: Verlag von R. Oldenbourg, 1951.
