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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?

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    $\begingroup$ While it's not a mathematical achievement as such, it is significant for mathematics that Oberwolfach was founded in 1944. $\endgroup$
    – stankewicz
    Commented Aug 19, 2010 at 17:17
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    $\begingroup$ There was some worthwhile applied mathematics happening at Bletchley Park (Enigma). Do you mean pure math unrelated to the war itself? $\endgroup$
    – KConrad
    Commented Aug 19, 2010 at 17:28
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    $\begingroup$ IIRC Turán's theorem on clique-free graphs was devised in a concentration camp. $\endgroup$
    – darij grinberg
    Commented Aug 19, 2010 at 19:59
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    $\begingroup$ More physics than math, but as I recall Krylov did fundamental work in theoretical statistical physics (specifically, he was primarily responsible for highlighting the role of mixing versus entropy) while serving in the Soviet artillery. $\endgroup$
    – Steve Huntsman
    Commented Aug 20, 2010 at 0:34
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    $\begingroup$ My recollection is that Turan's work obtained in forced labor camp (not concentration camp) was on the crossing number on complete bipartite graphs. His description of this is quoted on p 50 of "Geometric graphs and arrangements: some chapters from Combinational geometry" by Stefan Felsner available on books.google.com. Worth reading! $\endgroup$
    – Péter Komjáth
    Commented Nov 24, 2010 at 6:56

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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.

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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.

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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.

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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.

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    $\begingroup$ G. Hochschild's "On the Cohomology Groups of an Associative Algebra" (Annals of Mathematics 1945, the paper that founded Hochschild cohomology) addresses the author at "Aberdeen Proving Ground, Md.". It seems to me that this is by far not the only case of mathematicians working for the US military during and directly after WWII (although the Manhattan Project staff would hardly have put their locations on their publications). $\endgroup$
    – darij grinberg
    Commented Aug 19, 2010 at 20:03
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    $\begingroup$ At a conference 11 or 12 years ago, Saunders Mac Lane spoke briefly about -- as I recall -- he and Eilenberg "working on codes during the day and on cohomology of groups at night" during the war. $\endgroup$
    – Jeff Strom
    Commented Aug 21, 2010 at 13:40
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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).

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    $\begingroup$ Amazing ! Since WWII began in Dantzig (the city, today Gdansk). $\endgroup$
    – Denis Serre
    Commented Aug 30, 2012 at 9:49
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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.

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    $\begingroup$ According to Wikipedia: "During World War II [Selberg] worked in isolation due to the German occupation of Norway. After the war his accomplishments became known, including a proof that a positive proportion of the zeros of the Riemann zeta function lie on the [critical] line" $\endgroup$
    – Micah Milinovich
    Commented Aug 19, 2010 at 17:20
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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

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  • $\begingroup$ Turing has been mentioned more than once. $\endgroup$
    – Gerry Myerson
    Commented Jul 24, 2022 at 0:35
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Monte Carlo integration was first put to use during the Manhattan project.

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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.

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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.

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  • $\begingroup$ What is/was "Aberdeen Proving Ground"? $\endgroup$
    – Kevin H. Lin
    Commented Aug 21, 2010 at 11:46
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    $\begingroup$ army.mil/info/organization/apg $\endgroup$
    – Steve Huntsman
    Commented Aug 21, 2010 at 13:01
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    $\begingroup$ One would expect a Proving Ground to be a good place to do Mathematics (unless one realized that "proving" here means "testing"). $\endgroup$
    – Gerry Myerson
    Commented Jul 18, 2014 at 3:29
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    $\begingroup$ @Gerry: Ha! But I'd been to APG once upon a time and had a rather different impression--the ordnance was quite loud... $\endgroup$
    – Steve Huntsman
    Commented Jul 18, 2014 at 4:00
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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.

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    $\begingroup$ One should note that Hopf was only in Switzerland as he had to flee Nazi Germany. $\endgroup$
    – David Corwin
    Commented Aug 29, 2012 at 22:12
  • $\begingroup$ This is not exactly true, Hopf went to Zurich already in 1931. But it is true that he would have had to leave Germany after 1933 if he hadn‘t already done so. Germany actually withdraw his citizenship and confiscated his property in 1943. $\endgroup$
    – ThiKu
    Commented Dec 17, 2017 at 5:08
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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.

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    $\begingroup$ There is actually a note on this (althouhg in German) at ftp.mi.fu-berlin.de/usr/raut/gentzen/Gentzen.txt . $\endgroup$
    – darij grinberg
    Commented Aug 20, 2010 at 13:54
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    $\begingroup$ Wasn't Gentzen himself deeply involved in Nazism (true believer, party member) - and, in particular, wasn't he pro-Bieberbach? $\endgroup$
    – H A Helfgott
    Commented Mar 29, 2012 at 12:06
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    $\begingroup$ My understanding (I think from the collected works edited by Szabo) is that Gentzen was politically naive and only joined the Nazi party because someone told him it would be good for his career. He died of starvation in August 1945 in a Soviet prisoner of war camp - a huge tragedy for logic. (I also understand that several algebraists, students of Noether indeed, were much more enthusiastic Nazis, but survived the war.) $\endgroup$
    – Paul Taylor
    Commented Sep 1, 2021 at 19:04
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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.

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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.

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    $\begingroup$ @none, KConrad already alluded to that in a comment of 19 August, and OP indicated in reply that this was not the kind of thing he was looking for. $\endgroup$
    – Gerry Myerson
    Commented Dec 8, 2010 at 4:29
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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".

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    $\begingroup$ Hi Watson! Thanks for writing this. I hope you're well. $\endgroup$
    – David Hansen
    Commented Aug 9, 2011 at 3:01
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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.

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    $\begingroup$ exact result is a little bit overstatement. $\endgroup$
    – Denis Serre
    Commented Aug 30, 2012 at 9:45
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Onsager's solution of the 2-dimensional Ising model of ferromagnetism: https://en.wikipedia.org/wiki/Ising_model

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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.

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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,

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  • $\begingroup$ You can read this in his autobiography, in fact. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 17, 2017 at 6:38
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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.

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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.

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    $\begingroup$ What about WWII? $\endgroup$
    – Steve Huntsman
    Commented Aug 21, 2010 at 13:04
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    $\begingroup$ If we're talking about other wars, Takagi proved class field theory during WWI. $\endgroup$
    – David Corwin
    Commented Aug 29, 2012 at 22:12
  • $\begingroup$ In WWII there is much more regarding number theory and Turing. Some fiction about it is in the book trilogy Cryptonomicum $\endgroup$
    – Jesus Martinez Garcia
    Commented Sep 17, 2012 at 16:21
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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.

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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.

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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]

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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.

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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).

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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

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Ernst Witt claimed that he had discovered the Leech lattice in 1940, see e.g.

https://en.wikipedia.org/wiki/Leech_lattice

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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.

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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)

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    $\begingroup$ Leray, perhaps? See secure.wikimedia.org/wikipedia/en/wiki/Leray_spectral_sequence $\endgroup$
    – David Roberts ♦
    Commented Apr 10, 2011 at 22:33
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    $\begingroup$ I doubt it was Leray, since the French word for sheaf is faisceau which translates to "beam". I remember hearing a story that Norman Steenrod and someone else came up with the English words sheaf, stalk, and germ sitting on a front porch somewhere in the American midwest. I can't seem to find this story via google, but it seems possible since Steenrod was born in Dayton, Ohio. What do the French call stalks and germs? $\endgroup$
    – James Weigandt
    Commented Aug 29, 2012 at 20:06
  • $\begingroup$ @JamesWeigandt in French: stalk $=$ tige; germ $=$ germe $\endgroup$
    – YCor
    Commented Jul 23, 2022 at 7:29
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