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## Major mathematical advances past age fifty [closed]

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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Such questions are usually counted "community wiki". – Wadim Zudilin May 23 2010 at 7:01
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. – Pietro Majer May 23 2010 at 8:10
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.) – BCnrd May 23 2010 at 13:04
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. – paul Monsky Jun 1 2010 at 23:50
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"). – Victor Protsak Jun 2 2010 at 0:09

## closed as no longer relevant by Bill Johnson, Mark Meckes, Will Sawin, Felipe Voloch, Andres CaicedoAug 23 at 6:46

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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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. – Elemer E Rosinger Jul 3 2010 at 13:05

Uncle Petros proved Goldbach's conjecture just minutes before his death, when he was more than sixty.

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Burnside proved the $p^aq^b$ theorem at age 53.

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Fourier (1768 - 1830) presented his work Théorie analytique de la chaleur in 1822 at age 54.

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