Question

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?

Answer 1

Alan Turing is an obvious answer

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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,

Answer 6

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 teached set theory to his comrades while being at war, and this book were his lecture notes, so to say. He also gave his venia legendi lecture during the war while being on furlough.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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".

Answer 11

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.

http://en.wikipedia.org/wiki/Sheaf_(mathematics)

Answer 12

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.

Answer 13

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.

Answer 14

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 for Russell's affair, 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 fiction written about it.

Answer 15

Onsager's solution of the 2-dimensional Ising model of ferromagnetism: http://en.wikipedia.org/wiki/Ising_model

Answer 16

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.

Answer 17

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.

Answer 18

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.

Answer 19

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).

Answer 20

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.

Answer 21

Monte Carlo integration was first put to use during the Manhattan project.

Answer 22

In the opposite direction, here is an example (one of presumably hundreds of others) of work that got cut short by the war:

http://qjmath.oxfordjournals.org/cgi/pdf_extract/os-16/1/1

Read the dedication before the introduction.

Answer 23

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.

Answer 24

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. http://en.wikipedia.org/wiki/Alan_Turing

Answer 25

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.

Answer 26

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.