Notable mathematics during World War II

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?

Are you asking specifically about mathematics being done in harsh war conditions, or does the progress done through the war effort also count? — Thierry Zell, Aug 19 '10 at 16:56
I suppose I'm more interested in work done under war conditions, but either would be interesting. — David Hansen, Aug 19 '10 at 16:57
While it's not a mathematical achievement as such, it is significant for mathematics that Oberwolfach was founded in 1944. — stankewicz, Aug 19 '10 at 17:17
There was some worthwhile applied mathematics happening at Bletchley Park (Enigma). Do you mean pure math unrelated to the war itself? — KConrad, Aug 19 '10 at 17:28
Although it is not in the spirit of David's question, there is an article in the Amer. Math. Monthly on the development of applied math (in the USA) in World War II: M. Rees, "The Mathematical Sciences and World War II", 87 (1980), 607--621. A JSTOR link is http://www.jstor.org/stable/2320947. — KConrad, Aug 19 '10 at 18:28
IIRC Turán's theorem on clique-free graphs was devised in a concentration camp. — darij grinberg, Aug 19 '10 at 19:59
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. — Steve Huntsman, Aug 20 '10 at 00:34
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! — Péter Komjáth, Nov 24 '10 at 06:56
Arne Beurling did important work during World War II : He deciphered a german secret teletypewriter, see http://en.wikipedia.org/wiki/Arne_Beurling . — jjcale, Aug 29 '12 at 19:45
I once it heard it mentioned, in the presence of Armand Borel, that Leray invented spectral sequences in a prison camp. Borel audibly snorted and said "That was no prison camp. It was a country club." I deeply regret that I did not ask him to elaborate. — Steven Landsburg, Dec 21 '17 at 03:33
The Dachau concentration camp was establlished in 1933, and the extreme duress on academics of Jewish descent in Germany began in 1933 beginning with the passage of discriminatory laws. Read the Wiki bio on Lise Meitner for some history on the laws, their enforcement, and an example of remarkable research done under duress (including gender discrimination). // Perhaps the Scottish Book, compiled in Poland from 1935-41, is an example of notable math done during the war. — Tom Copeland, Nov 17 '21 at 10:03

score 40 · Answer 1 · edited Aug 29 '12 at 14:46

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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.

edited Aug 29 '12 at 14:46

answered Aug 19 '10 at 18:44

Greg Kuperberg

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score 28 · Answer 2 · edited Dec 20 '17 at 13:03

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.

score 26 · Answer 3 · edited Jul 03 '22 at 09:08

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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.

edited Jul 03 '22 at 09:08

Glorfindel

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answered Aug 19 '10 at 20:23

user12345678

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KConrad · Answer 4 · 2010-08-19T19:06:03.703

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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.

edited Aug 19 '10 at 19:06

answered Aug 19 '10 at 16:56

KConrad

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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). – darij grinberg Aug 19 '10 at 20:03
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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. – Jeff Strom Aug 21 '10 at 13:40

score 21 · Answer 5 · answered Aug 19 '10 at 22:08

21

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).

answered Aug 19 '10 at 22:08

J. M. isn't a mathematician

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Amazing ! Since WWII began in Dantzig (the city, today Gdansk). – Denis Serre Aug 30 '12 at 09:49

score 20 · Answer 6 · answered Aug 19 '10 at 17:05

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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.

answered Aug 19 '10 at 17:05

Micah Milinovich

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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" – Micah Milinovich Aug 19 '10 at 17:20

score 19 · Answer 7 · edited Jul 23 '22 at 05:32

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

Turing has been mentioned more than once. – Gerry Myerson Jul 24 '22 at 00:35 — Gerry Myerson, Jul 24 '22 at 00:35

score 16 · Answer 8 · answered Aug 19 '10 at 20:12

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

answered Aug 19 '10 at 20:12

user1504

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score 13 · Answer 9 · edited Aug 21 '10 at 07:29

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.

score 12 · Answer 10 · edited Jul 23 '22 at 05:31

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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.

edited Jul 23 '22 at 05:31

Martin Sleziak

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answered Aug 20 '10 at 00:18

Steve Huntsman

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What is/was "Aberdeen Proving Ground"? – Kevin H. Lin Aug 21 '10 at 11:46
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http://www.army.mil/info/organization/apg/ – Steve Huntsman Aug 21 '10 at 13:01
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One would expect a Proving Ground to be a good place to do Mathematics (unless one realized that "proving" here means "testing"). – Gerry Myerson Jul 18 '14 at 03:29
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@Gerry: Ha! But I'd been to APG once upon a time and had a rather different impression--the ordnance was quite loud... – Steve Huntsman Jul 18 '14 at 04:00

score 10 · Answer 11 · answered Aug 09 '11 at 20:16

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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.

answered Aug 09 '11 at 20:16

Alain Valette

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One should note that Hopf was only in Switzerland as he had to flee Nazi Germany. – David Corwin Aug 29 '12 at 22:12
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. – ThiKu Dec 17 '17 at 05:08

score 9 · Answer 12 · answered Aug 20 '10 at 13:47

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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.

answered Aug 20 '10 at 13:47

Stefan Hoffelner

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There is actually a note on this (althouhg in German) at http://ftp.mi.fu-berlin.de/usr/raut/gentzen/Gentzen.txt . – darij grinberg Aug 20 '10 at 13:54
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Wasn't Gentzen himself deeply involved in Nazism (true believer, party member) - and, in particular, wasn't he pro-Bieberbach? – H A Helfgott Mar 29 '12 at 12:06
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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.) – Paul Taylor Sep 01 '21 at 19:04

score 7 · Answer 13 · edited Jul 23 '22 at 05:29

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

edited Jul 23 '22 at 05:29

Martin Sleziak

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answered Aug 20 '10 at 16:14

Michael Thaddeus

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score 7 · Answer 14 · answered Dec 07 '10 at 16:17

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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.

answered Dec 07 '10 at 16:17

Denis Serre

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score 7 · Answer 15 · answered Dec 08 '10 at 02:12

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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.

answered Dec 08 '10 at 02:12

none

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@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. – Gerry Myerson Dec 08 '10 at 04:29

score 7 · Answer 16 · answered Aug 09 '11 at 02:54

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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".

answered Aug 09 '11 at 02:54

Watson Ladd

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Hi Watson! Thanks for writing this. I hope you're well. – David Hansen Aug 09 '11 at 03:01

score 7 · Answer 17 · answered Aug 09 '11 at 06:46

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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.

answered Aug 09 '11 at 06:46

Andrew

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exact result is a little bit overstatement. – Denis Serre Aug 30 '12 at 09:45

score 6 · Answer 18 · answered Aug 30 '12 at 00:45

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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.

answered Aug 30 '12 at 00:45

none

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score 4 · Answer 19 · answered Aug 29 '12 at 17:28

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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,

answered Aug 29 '12 at 17:28

jbc

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You can read this in his autobiography, in fact. – Mariano Suárez-Álvarez Dec 17 '17 at 06:38

score 4 · Answer 20 · answered Aug 29 '12 at 18:12

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.

score 4 · Answer 21 · edited Jul 23 '22 at 05:28

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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.

edited Jul 23 '22 at 05:28

Martin Sleziak

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answered Aug 21 '10 at 08:18

Jesus Martinez Garcia

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What about WWII? – Steve Huntsman Aug 21 '10 at 13:04
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If we're talking about other wars, Takagi proved class field theory during WWI. – David Corwin Aug 29 '12 at 22:12
In WWII there is much more regarding number theory and Turing. Some fiction about it is in the book trilogy Cryptonomicum – Jesus Martinez Garcia Sep 17 '12 at 16:21

score 3 · Answer 22 · edited Dec 19 '17 at 20:34

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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.

edited Dec 19 '17 at 20:34

Ben McKay

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answered Aug 29 '12 at 14:38

Wolfgang Jeltsch

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score 3 · Answer 23 · answered Dec 19 '17 at 17:47

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.

score 3 · Answer 24 · answered Nov 13 '21 at 05:10

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]

score 2 · Answer 25 · answered Aug 09 '11 at 17:20

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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.

answered Aug 09 '11 at 17:20

Michael Greenblatt

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score 1 · Answer 26 · edited Aug 29 '21 at 19:07

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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).

edited Aug 29 '21 at 19:07

მამუკა ჯიბლაძე

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answered Dec 22 '17 at 06:36

Jules Wei

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score 1 · Answer 27 · answered Aug 29 '21 at 21:48

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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

answered Aug 29 '21 at 21:48

abo

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score 1 · Answer 28 · answered Aug 30 '21 at 10:44

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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

answered Aug 30 '21 at 10:44

Martin Seysen

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score 1 · Answer 29 · answered Aug 30 '21 at 13:08

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.

score 1 · Answer 30 · edited Jul 23 '22 at 05:27

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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)

edited Jul 23 '22 at 05:27

Martin Sleziak

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answered Apr 10 '11 at 14:24

Daniel Parry

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Leray, perhaps? See https://secure.wikimedia.org/wikipedia/en/wiki/Leray_spectral_sequence – David Roberts Apr 10 '11 at 22:33
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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?
– James Weigandt Aug 29 '12 at 20:06
@JamesWeigandt in French: stalk $=$ tige; germ $=$ germe – YCor Jul 23 '22 at 07:29

score 0 · Answer 31 · answered Aug 29 '12 at 22:16

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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.

answered Aug 29 '12 at 22:16

David Corwin

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As noted in the first sentence of the question, right? – Gerry Myerson Aug 29 '12 at 23:44
See my relevant comment on the question. – Steven Landsburg Dec 21 '17 at 03:33

score 0 · Answer 32 · answered Aug 30 '12 at 09:26

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Alan Turing is an obvious answer

answered Aug 30 '12 at 09:26

RougeSegwayUser

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Already mentioned twice – Sune Jakobsen Aug 30 '12 at 10:42

score 0 · Answer 33 · answered Aug 29 '21 at 23:01

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 $r$).

score 0 · Answer 34 · answered Aug 30 '21 at 02:21

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.

score 0 · Answer 35 · answered Nov 13 '21 at 13:02

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.

Notable mathematics during World War II

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