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From A Mathematician’s Apology, G. H. Hardy, 1940: "I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. ... I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself."

Have matters improved for the elderly mathematician? Please answer with major discoveries made by mathematicians past 50.

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    $\begingroup$ Rmk: Hardy suffered of depression, and was living not exactly in the most suitable environment for that. Unfortunately, this wrong idea of "mathematics is a young man's game" had an incredible success. $\endgroup$
    – Pietro Majer
    Commented May 23, 2010 at 8:10
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    $\begingroup$ Cliff Taubes (b. 1954) recently solved Weinstein conjecture, Gopal Prasad (b. 1945) has done multiple great things (separately with J-K. Yu, A. Rapinchuk, & S-K. Yeung) on buildings, Zariski-dense and arithmetic subgroups of ss groups over number fields, classification of "fake" projective spaces, etc., Serre turned 50 in 1976 (e.g., his precise modularity conjecture published in 1986 exerted vast influence over number theory ever since), and Jean-Marc Fontaine (b. 1944) is as dominant as ever in $p$-adic Hodge theory (e.g., Colmez-Fontaine thm. in 2000, recent work with L. Fargues, etc.) $\endgroup$
    – BCnrd
    Commented May 23, 2010 at 13:04
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    $\begingroup$ This isn't exactly what you were asking for, but Littlewood himself, after overcoming depression at age 72, did good mathematics throughout his 80's--it's hardly a young man's game. $\endgroup$
    – paul Monsky
    Commented Jun 1, 2010 at 23:50
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    $\begingroup$ Re "Littlewood himself": Of course it was well known that Littlewood was the name Hardy used to publish his lesser results (cf "A mathematician's miscellany"). $\endgroup$
    – Victor Protsak
    Commented Jun 2, 2010 at 0:09
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    $\begingroup$ What's really odd is when Abel and Galois are wheeled out in support of the view that mathematics is a young person's game. Spot the logical flaw. $\endgroup$
    – Tom Leinster
    Commented Aug 20, 2012 at 18:25

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Poincaré's conjecture has been formulated in 1904, when he had just turned 50, while presenting a counter-example (the Poincaré homology sphere) to another earlier conjecture of his. Probably, given the impact it has had for a whole century, the precise formulation of the conjecture can be seen as a "major discovery" by itself.

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    $\begingroup$ Just for the record, Poincaré never actually expressed his so-called conjecture as a conjecture; rather he brought it up as a question. After incorrectly making the conjecture that homology suffices to detect a 3-sphere -- and ingeniously finding a counterexample to that -- he was evidently chastened enough to refrain from phrasing what is called the Poincaré Conjecture as an actual conjecture. $\endgroup$
    – Daniel Asimov
    Commented May 25, 2010 at 1:50
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    $\begingroup$ By the way, to get the accents in comments you can just copy-paste: Poincaré. $\endgroup$
    – Victor Protsak
    Commented May 25, 2010 at 4:46
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Burnside proved the $p^aq^b$ theorem at age 53.

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George Pólya (1887-1985) wrote the wonderful paper

Pólya, George, On the eigenvalues of vibrating membranes, Proc. Lond. Math. Soc., III. Ser. 11, 419-433 (1961). ZBL0107.41805.

at the age of 73. This paper motivated large chunk of research known as the Pólya conjecture of the eigenvalues of the Laplacian. See for example this MO-Question.

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Charles Sanders Peirce (born 1839) explicitly declared his Existential Graphs (all three parts: Alpha, Beta, and Gamma) to be his chef d'oeuvre. This work on graphical logic began sometime in the early 1880's, and he continued to work on it until his death in 1914.

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  • $\begingroup$ For the lazy, (1880,1914)-1839=(41,75). $\endgroup$
    – Federico Poloni
    Commented Oct 27, 2013 at 12:46
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Uncle Petros proved Goldbach's conjecture just minutes before his death, when he was more than sixty.

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  • $\begingroup$ Can we see the proof? Gerhard "I'm (not) From Missouri, Mister" Paseman, 2011.02.15 $\endgroup$
    – Gerhard Paseman
    Commented Feb 16, 2011 at 0:59
  • $\begingroup$ C'mon, people: where's your sense of fun here? :-) $\endgroup$
    – Todd Trimble
    Commented Aug 20, 2012 at 21:49
  • $\begingroup$ @Todd: it died once I realized I'd probably never get away with helping John Rainwater write up some more "folklore made explicit". $\endgroup$
    – Yemon Choi
    Commented Aug 20, 2012 at 23:28
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The story with one's age is very simple : different persons can age very differently. If one takes care not to age in the wrong way for a given intellectual venture, then quite likely, one can pursue it for many decades ... And of course, mathematics is an intellectual venture ... A good example of how little physical condition is needed for pursuing an intellectual venture is given by the well known physicist Stephen Hawking ...

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    $\begingroup$ By the way of mathematics, the Austrian mathematician Leopold Vietoris (4 June 1891 – 9 April 2002) has published papers till his last days. And after retirement, he published more than during his academic career. $\endgroup$
    – Elemer E Rosinger
    Commented Jul 3, 2010 at 13:05
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Fourier (1768 - 1830) presented his work Théorie analytique de la chaleur in 1822 at age 54.

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Mihailescu https://en.wikipedia.org/wiki/Preda_Mih%C4%83ilescu (born 1955) who proved the https://en.wikipedia.org/wiki/Catalan%27s_conjecture in 2002.

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  • $\begingroup$ Really interesting. Thanks Timo. (+7) years upper than bound. $\endgroup$
    – user36136
    Commented Oct 27, 2013 at 11:43
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    $\begingroup$ For the lazy, 2002-1955=47. $\endgroup$
    – Federico Poloni
    Commented Oct 27, 2013 at 12:45
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    $\begingroup$ this answer, though great, already appeared here 3 years ago $\endgroup$
    – András Bátkai
    Commented Oct 27, 2013 at 21:53
  • $\begingroup$ @AndrásBátkai, note that this answer was transferred here from another, newer, question, where it had not yet appeared. $\endgroup$
    – Gerry Myerson
    Commented Oct 27, 2013 at 22:31
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I think one of the best examples is "Abraham Robinson" who made many important contributions after his 40th. I even read somewhere (I don't remember where) that he was very happy for this.

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  • $\begingroup$ Thanks. Can you specify one of Robinson's works as his masterwork after 40 years old? $\endgroup$
    – user36136
    Commented Oct 27, 2013 at 12:03
  • $\begingroup$ For example, the creation of non-standard analysis. $\endgroup$
    – Mohammad Golshani
    Commented Oct 27, 2013 at 12:05
  • $\begingroup$ Great! It is a real masterwork. $\endgroup$
    – user36136
    Commented Oct 27, 2013 at 12:09
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Grothendieck wrote "Pursuing Stacks", aka his letter to Quillen, at roughly the age of 55.

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Ludolph van Ceulen was 56 when he published 20 digits of $\pi$, and he later expanded this to 35 digits. He was appointed as a professor when he was 60. Computing 35 digits of $\pi$ may seem easy now, but he did it before calculus. In some German universities $\pi$ was called "Ludolph's number" even into the 20th century.

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