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.

• Such questions are usually counted "community wiki". – Wadim Zudilin May 23 '10 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 '10 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 '10 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 '10 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 '10 at 0:09

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.

• Just for the record, Poincar&eacute; 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 Poincar&eacute; Conjecture as an actual conjecture. – Daniel Asimov May 25 '10 at 1:50
• By the way, to get the accents in comments you can just copy-paste: Poincaré. – Victor Protsak May 25 '10 at 4:46

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

• Can we see the proof? Gerhard "I'm (not) From Missouri, Mister" Paseman, 2011.02.15 – Gerhard Paseman Feb 16 '11 at 0:59
• C'mon, people: where's your sense of fun here? :-) – Todd Trimble Aug 20 '12 at 21:49
• @Todd: it died once I realized I'd probably never get away with helping John Rainwater write up some more "folklore made explicit". – Yemon Choi Aug 20 '12 at 23:28

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

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

• Really interesting. Thanks Timo. (+7) years upper than bound. – user36136 Oct 27 '13 at 11:43
• For the lazy, 2002-1955=47. – Federico Poloni Oct 27 '13 at 12:45
• this answer, though great, already appeared here 3 years ago – András Bátkai Oct 27 '13 at 21:53
• @AndrásBátkai, note that this answer was transferred here from another, newer, question, where it had not yet appeared. – Gerry Myerson Oct 27 '13 at 22:31

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.

• For the lazy, (1880,1914)-1839=(41,75). – Federico Poloni Oct 27 '13 at 12:46

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 ...

• 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 '10 at 13:05

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

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.

• Thanks. Can you specify one of Robinson's works as his masterwork after 40 years old? – user36136 Oct 27 '13 at 12:03
• For example, the creation of non-standard analysis. – Mohammad Golshani Oct 27 '13 at 12:05
• Great! It is a real masterwork. – user36136 Oct 27 '13 at 12:09