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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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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.

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    $\begingroup$ As noted in the first sentence of the question, right? $\endgroup$
    – Gerry Myerson
    Commented Aug 29, 2012 at 23:44
  • $\begingroup$ See my relevant comment on the question. $\endgroup$
    – Steven Landsburg
    Commented Dec 21, 2017 at 3:33
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Alan Turing is an obvious answer

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    $\begingroup$ Already mentioned twice $\endgroup$
    – Sune Jakobsen
    Commented Aug 30, 2012 at 10:42
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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 $r$).

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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.

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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.

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