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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.

Sam Nead
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Halfdan Faber
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    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 08:10
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    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
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    "Please answer with major discoveries made by mathematicians past 50." Of course, if you specify the format for an admissible answer in such a way, you can get any result you want. How about "Answer with a name of a good mathematician who hasn't done anything of importance after the age 50?" instead? (Well, let's be fair and restrict to those who lived until at least 60). – fedja May 23 '10 at 19:26
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    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 01 '10 at 23:50
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    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 02 '10 at 00:09
  • See: http://sps.nus.edu.sg/~limchuwe/articles/youth.html – Halfdan Faber Jun 26 '10 at 20:48
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    This BETTER be a myth given how late I started out. But there's plenty of evidence it is. – The Mathemagician Jul 19 '10 at 05:59
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    I suspect that looking for published papers, clearly attributed, undersamples the contributions of older mathematicians. They publish less, on average, because they are far more likely to have tenure. This doesn't mean that they don't have new ideas, but that their big ideas are more likely to be disseminated by students and collaborators. – user1504 Feb 15 '11 at 15:14
  • Michael Aschbacher (born 1944) and Stephen D. Smith (born 1948) published the classification of quasithin groups - the missing link in the classification of finite simple groups - in 2004. – Someone Jul 20 '11 at 15:08
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    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. – Tom Leinster Aug 20 '12 at 18:25
  • just know some probability mathematicians like Dynkin who are alive at 90+ and working. if they are great so must be producing some good stuff? –  Oct 27 '13 at 11:29
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    carleson's work on henon map in his seventies – Koushik Dec 12 '13 at 11:31
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    In evaluating Hardy's claim today, it should absolutely be taken into account that mathematics has become a highly competitive field where you need substantial early success to advance. So I think it's more than reasonable to assume there is some survivorship bias ― a person whose best ideas come later in life may not make it to that career stage at all. – R. van Dobben de Bruyn Mar 23 '22 at 13:17

41 Answers41

71

Roger Apery was 62 when he proved the irrationality of $\zeta(3)$.

David Hansen
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    +1. And the related link: http://peccatte.karefil.com/PhiMathsTextes/Apery.html – Wadim Zudilin May 23 '10 at 08:37
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    I remember reading that Apéry was snubbed by professional mathematical community and his proof was disregarded at first. – Victor Protsak May 23 '10 at 10:02
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    I think disregarded is a bit too strong - more like, met with considerable skepticism, on the grounds that it was not expected that such a venerable problem would be solved by such low-tech methods. The community took the proof seriously enough to go through it in detail and then acknowledged that it was valid. – Gerry Myerson May 23 '10 at 10:22
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    Victor, read the article "A proof that Euler missed" http://www.maths.mq.edu.au/~alf/45.pdf This provides some historical context around the announcement of Apery's proof. Skepticism toward proofs the prover has not produced the details of is healthy, not snubbing. – Ben Webster May 23 '10 at 14:08
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    Thank you for the link to van der Poorten's article, Ben! Upon re-reading it and the article that Wadim linked it became clear that the so-called "community" acted in a worst possible manner. It was only thanks to the determination of a few outstanding mathematicians that he got the recognition that he deserved. – Victor Protsak May 23 '10 at 23:59
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    van der Poorten: 1. "Though there had been earlier rumours of his claiming a proof, scepticism was general." 2. "I heard with some incredulity that, for one, Henri Cohen (then Bordeaux, now Grenoble) believed that these claims might well be valid." 3. "We were quite unable to prove that the sequences $(a_n)$ defined above did satisfy the recurrence (1.2) (Apéry rather tartly pointed out to me in Helsinki that he regarded this more a compliment than a criticism of his method)." – Victor Protsak May 24 '10 at 00:07
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    van der Poorten: 4. "Anyhow, I considered ‘A Proof that Euler Missed’ a racy title. It arose after Cohen’s report at Helsinki, with someone (specifically, an old friend I was sitting next to) sourly commenting ‘A victory for the French peasant...’. To this Nick Katz retorted: ‘No! No! This is marvellous! It is something Euler could have done.’" – Victor Protsak May 24 '10 at 00:08
  • François Apéry: 1. "The dominance of Bourbaki meant marginalization for the anti-Bourbakiste. Not being in sympathy even with all the other marginalized, Apéry eventually found himself nearly isolated." – Victor Protsak May 24 '10 at 00:17
  • François Apéry: 2. "The instigators of the Lichnérowicz reform insisted on loyalty to their program and tried to brand any opposition to it as reactionary, which only hardened Apéry’s position and deepened his isolation in the community. It went so far that at the Journées Arithmétiques de Marseille in 1978, his lecture on the irrationality of z (3) was greeted with doubt, disbelief, and then disorder. Its recognition at the Helsinki Congress would finally erase this humiliation." – Victor Protsak May 24 '10 at 00:18
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    "Upon re-reading it and the article that Wadim linked it became clear that the so-called "community" acted in a worst possible manner. It was only thanks to the determination of a few outstanding mathematicians that he got the recognition that he deserved."

    I would say this differently. Thru the determination of others Apery's ideas went from a convoluted multi-100 page work to a 3-page note. Has anyone bothered to see if his original manuscript did in fact prove bounded denominators? That's the crux, and vDP's cheeky "utterly compelling" numerically (so is 1.2020569..., no?) is a nip off.

     – Junkie May 24 '10 at 02:43
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    Henri Cohen told me that he and Hendrik Lenstra were in the audience at one of Apery's first public lectures about his proof. He said that they were madly scribbling during the talk, and by the end they were convinced. – Victor Miller May 24 '10 at 03:39
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    "Cohen told me that he and Lenstra were in the audience at one of Apery's first public lectures about his proof. He said that they were madly scribbling during the talk, and by the end they were convinced." I heard Lenstra did real-time numerics on his calculator to verify some claims, and gave them greater confidence to delve further. As vdP says, they realised the crux later (despite being "convinced" previously), and was unproved (to them) prior to Zagier. I dont know if Apery stressed where the difficulties were, or if he was asked how it worked. A=B for identities wasnt known widely then – Junkie May 24 '10 at 10:50
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    The following is my (probably flawed) recollection of part of Cohen's lecture at the Lenstra Truerfeest a little over 7 years ago: "Apéry gave a shameful talk. He explained almost nothing, and many of his formulas didn't make any sense. One of his sums seemed to have zeroes in the denominator of every term. But there was one formula that he wrote that looked interesting and new, and Hendrik was sitting next to me with a calculator. I asked him to check the first few terms on his calculator, and they matched very well. After the talk we were able to use it to reconstruct a proof." – S. Carnahan May 25 '10 at 03:36
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    I think between "Hendrik was sitting next to me with a calculator." and "I asked him to check the first few terms", Cohen made some remark about calculators being rare and expensive back then. – S. Carnahan May 25 '10 at 03:49
57

Since no one has mentioned A.N. Kolmogorov (born 1903), I hope I may be forgiven for a second answer. The following is from Kolmogorov's Wikipedia biography.

In classical mechanics, he is best known for the Kolmogorov–Arnold–Moser theorem (first presented in 1954 at the International Congress of Mathematicians). In 1957 he solved Hilbert's thirteenth problem (a joint work with his student V. I. Arnold). He was a founder of algorithmic complexity theory, often referred to as Kolmogorov complexity theory, which he began to develop around this time.

John Stillwell
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    John, Great answer! I don't think there should be any limit on the number of good answers from any one contributor, for a question like this. – Halfdan Faber May 25 '10 at 07:47
55

Weierstrass approximation theorem was proved by Karl Weierstrass when he was 70 years old

arun s
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54

An answer of particular contemporary relevance would be Yitang Zhang, who established earlier this year (2013) that there are infinitely many pairs of primes which differ by less than 70 million (this constant has subsequently been improved to about 5,000). He was born in 1955 and had only two previous journal publications.

Ian Morris
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52

Kurt Heegner published his only, extremely influential paper, in 1952 when he was 59. However it took nearly 20 years for the mathematical community to realize what a gem it was.

Victor Miller
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Leonhard Euler. According to the Wikipedia page, he still managed to produce one paper per week in the year 1775 (at age 68), despite deteriorating eyesight. As a concrete example, at age 65 he proved that $2^{31} − 1$ is a Mersenne prime, which may have remained the largest known prime for the next 95 years.

Martin Sleziak
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Tony Huynh
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  • A more precise version of the history on M_31 can be found here: http://primes.utm.edu/notes/by_year.html He certainly did not do the necessary calculations himself at a time when he was completely blind. – Franz Lemmermeyer May 23 '10 at 10:58
47

Marina Ratner (b. 1938) proved Ratner's Theorems around 1990. They are some of the biggest advances in ergodic theory for quite a long time.

jeebus
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P. S. Novikov was 54 when he gave the first proof (143 pages!) of the unsolvability of the word problem for groups in 1955, and 58 when he co-solved the Burnside problem with S. I. Adian.

John Stillwell
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  • But Sergey Ivanovich (Adian) was much younger at the time of solving the Burnside problem! +1 for recalling a remarkable family of results. – Wadim Zudilin May 23 '10 at 12:46
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    Another in the same family of results is the theorem of A. A. Markov that the homeomorphism problem is unsolvable for manifolds of dimension $\ge 4$, proved in 1958 when Markov was 55. – John Stillwell May 23 '10 at 23:17
42

There are many examples of people doing significant work into their 60s and 70s, but fewer great discoveries. Here are a couple of my favorites:

  1. August Ferdinand Möbius discovered the Möbius band in 1858 at age 68 (the date referenced in Wikipedia). Other sources place the discovery even later: in 1861 he submitted to the French Academy prize competition a paper on it that passed unnoticed. As John Stillwell pointed out, in 1863 (age 73), Möbius published the classification of surfaces by genus (and in 1865 he finally described the Möbius band and the notion of orientability in print). Johann Benedict Listing turned 54 in 1862, the year in which he published a memoir discussing a 4-dimensional generalization of Euler's formula and described the Möbius band which he discovered independently.

  2. Julius Plücker was 64 in 1865, when he "returned to the field of geometry" after a hiatus of nearly 20 years (Wikipedia, McTutor, Cajori) and discovered the "line geometry" (it is possible that the roots of this discovery go back to his 1846 monograph). The first volume of his book Neue Geometrie des Raumes describing it was published in 1868 and the second volume was completed and published posthumously by Felix Klein in 1869. The idea of using higher-dimensional objects as points in new "geometry" made profound impact on Klein and Sophus Lie and led to the Erlangen program and, by route of Lie sphere geometry, to Lie's general theory of transformation groups. This also marked one of the first appearances of higer-dimensional spaces in geometry.

The question has been closed, but perhaps the following recent example deserves mention:
  1. As described in a Quanta magazine article, a retired German statistician Thomas Royen proved the Gaussian correlation inequality (GCI) in 2014 at the age of 67. GCI was a major conjecture at the interface of probability and convex geometry that remained open for more than 40 years. An additional twist to the story is that the proof went virtually unnoticed for almost 2 years.
Martin Sleziak
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Victor Protsak
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    At an even later age, Möbius in 1863 discovered the classification of closed orientable surfaces by genus. – John Stillwell May 23 '10 at 08:51
  • Thank you, I didn't remember that! On the other hand, Kolmogorov and Yushkevich ("Mathematics of the 19th century", vol 2) indicate that Plücker's idea of line geometry originated already in his 1846 "System des Geometrie des Raumes". I don't have Klein's "Lectures on the development of mathematics in the 19th century" close at hand to clarify this point, but apparently Plücker switched to doing physics around that time (1846) due to strained relations with German mathematicians and unfavorable reception of his analytic methods. – Victor Protsak May 23 '10 at 09:13
  • "The idea of using higher-dimensional objects as points" - What does this mean? Could it mean that Plucker's idea led to generic points in algebraic geometry (and more generally, non-closed points of schemes)? – David Corwin Aug 21 '12 at 01:47
  • No, just to the idea of more abstract spaces, such as homogeneous spaces $G/H$ and the simplest examples of moduli spaces. – Victor Protsak Aug 23 '12 at 01:34
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According to wiki, Mihailescu got his PhD at the age of 42; and then proved Catalan's conjecture in 2002, age 47, so almost 50.

Martin Sleziak
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brondell
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Zariski proved what might be arguably his greatest result, the theorem on formal functions, just after turning fifty. He also initiated a whole field of enquiry, the theory of equisingularity, in his late 60's.

Angelo
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Although I concede that there is some truth to the belief that the greatest conceptual breakthroughs in mathematics are made by younger mathematicians, I think it has led to the mistaken idea that older mathematicians rarely do anything significant.

I just don't think it's that uncommon for top mathematicians today to be productive after they're 50. Atiyah and Bott did great work after they were 50. It seems to me that so did Singer. Although most mathematicians slow down after they are 50, so do most non-mathematicians. But there are not a few exceptions to this.

And is any of this that different from other fields?

Deane Yang
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    +1 for the last line especially. – Mark Meckes May 23 '10 at 18:24
  • He told me several anecdotes about Hardy, but he presented each story in a sarcastic tone. “Hardy’s opinion that mathematics is a young man’s game is nonsense,” he said. (Goro Shimura, André Weil As I Knew Him, http://www.ams.org/notices/199904/shimura.pdf) – Victor Protsak May 24 '10 at 01:37
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    How are other fields different? Well, I always picture experimental sciences as being much more hierarchical, with the (older) team leader being credited for the work of the whole team. Thus, one's impressive achievements would come later in life. Disclaimer: I don't know what I'm talking about, as I wrote it's the image I got, but not necessarily from very reliable sources. – Thierry Zell Feb 15 '11 at 16:56
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Louis de Branges solved the Bieberbach conjecture in 1985 when he was 53.

Victor Miller
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  • Was that his first or his last attempt at the solution? I understand that he worked on it for quite some time. – Victor Protsak May 23 '10 at 22:41
  • Depending on how one counts it might have been his second or third. de Branges is no shrinking violet. He's also announced solutions to the Poincare conjecture (before Perlman) and the Riemann Hypothesis. – Victor Miller May 24 '10 at 03:36
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    I'm not aware that de Branges ever claimed a proof of the Poincare conjecture, which is a bit far from de Branges' area of research. He has been working relentlessly on the Riemann hypothesis for quite a while now. – Deane Yang May 24 '10 at 15:25
  • @Deane A few months ago,he claimed at the archive to actually have the proof-but so far,no one's been able to verify it. I hope someone really takes a good hard look at it-if only to stop him from saying it. – The Mathemagician Jul 19 '10 at 06:10
  • @VictorMiller I suspect you're confusing the Poincare conjecture with the invariant subspace problem. Karl Sabbagh quotes de Branges as saying, "The first case in which I made an error was in proving the existence of invariant subspaces for continuous transformations in Hilbert spaces. This was something that happened in 1964, and I declared something to be true which I was not able to substantiate. And the fact that I did that destroyed my career. My colleagues have never forgiven it." – Timothy Chow Nov 11 '21 at 17:39
  • @TimChow You may be right. I only heard this about de Branges from some mathematicians I knew at IBM. This was a long time ago ... – Victor Miller Nov 12 '21 at 18:11
25

Theorema Egregium was published by Gauss in 1828. Since Gauss was born in 1777, he ought to have been a little over 50 then.

Ref: Disquisitiones generales circa superficies curva (1828)

Martin Sleziak
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Anweshi
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    Gauss usually keeps his stuff in his sleeve for a while. So you never knew when he had the insight. – Turbo Aug 21 '12 at 04:20
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Furtwängler proved the principal ideal theorem when he was almost 60. No small feat given that Artin and Schreier simultaneously were working on it.

Franz Lemmermeyer
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Paul Erdős continued to do work in many fields including combinatorics after his 50th birthday. Some of his papers are here

Martin Sleziak
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Kristal Cantwell
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20

Something fitting this description that I haven't seen mentioned here is Norman Levinson's proof that asymptotically 1/3 of the zeroes of the Riemann zeta function lie on the critical line, which was the best result of its kind at the time. He was a little over 60 when he proved this, shortly before his death. What I find most remarkable about this is that he didn't really do much number theory until his last few years.

Michael Greenblatt
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Christos Papadimitriou is in his late 50's now (I can't find his exact age, which is a little strange), and in just the past few years he's done major work in algorithmic game theory, a field at least somewhat removed from the one he made his career in. Technically, he's a theoretical computer scientist - I say this is close enough though.

DoubleJay
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This is not really an answer but an objection to most of the answers at this pages and in particular to not so well formed question (it does not do justice to Hardy's book in my opinion).

If you read the whole chapter of Hardy's book where the excerpt is from, Hardy explains somewhere that he does not know a highest class mathematician whose best discoveries came after 50. I recall after reading the whole chapter that I was convinced with the bulk of text that Hardy meant that there are no major advances by a mathematician after 50, unless they had major discoveries also before 50. So Euler and Poincare are not counterexamples to Hardy's experience, and some other answers in this column are not as well! Of course some people completed earlier work after 50, or continued with major advances while they already became major mathematicians before, but do you really a know a mathematician who done no major research before 50 and done such world class research after 50 ?? Also do not look the publication dates but the creation dates.

Zoran Skoda
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    well, even your formulation is a bit extreme. I don't see the need to require "no major research before 50". How about just a well-known mathematician whose best work was after the age of 50? Although this is less common, I don't think it is any more rare than it is in any other field. – Deane Yang May 24 '10 at 14:38
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    As I understand it, Apery, who has already been mentioned, remains a good example even if you restrict the question in this way. – gowers May 25 '10 at 12:54
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    @Deane,gowers: Thank you,gentleman.This old wives' tale of Hardy's has probably prevented many a late bloomer from pursuing thier dreams. And that's tragic. – The Mathemagician Jul 19 '10 at 06:04
  • @Deane: "How about just a well-known mathematician whose best work was after the age of 50?" Examples? How about a man who began his mathematical work (or even became a mathematician) after 50 ?! – Sergei Tropanets Jul 19 '10 at 13:27
  • 50 is pretty high related to the old saw from physics where "if you haven't done great work by the age of 28, you might as well give up." – Viktor Bundle Jun 25 '11 at 02:59
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    Andrew, a person who does mathematics in order to be famous or "major mathematician" and not to enjoy path of curiosity is tragic in the first place, whatever be his/her achievements. The present day celebrity counterculture, statistics based on "official results", meaningless linear lists of comparison and so on aggravate the situation. Among the worst is the funding agency habit that they award more future support to those who got in past more support, else param. equal. That is, if you achieved the same with more spending in past they consider you more efficient, what is outrageously wrong. – Zoran Skoda Jun 26 '11 at 08:59
  • Yes, as @Zoran observes, highly biased "estimators"... :) – paul garrett Aug 20 '12 at 23:18
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    People who do what is clearly their best work (in the strict sense, not just "work that is as good as what they did before") around 50 or later do exist - see, e.g., Marina Ratner (what would be some other indisputable examples?) - but they are rare - rare enough that people remark repeatedly on it. – H A Helfgott Jan 06 '22 at 16:32
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Philip Hall published his paper with Higman, as well as his "Theorems like Sylow's", after he was 50. These are arguably his two biggest papers (and the Hall-Higman paper is arguably one of the most important papers in group theory).

Steve

Steve D
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And of course, Dennis Sullivan and James Stasheff, both well into their 60's and 70's, are still both major contributors to topology and categorical algebra.

Martin Sleziak
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The Mathemagician
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Tibor Rado introduced the busy beaver function and proved its noncomputablity at the age of 67.

Péter Komjáth
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Karl Dickman (born 1862) published the only math paper in 1930 (age 68) about distribution of prime factors. He discovered the asymptotic distribution of the largest prime divisor of n, where n is chosen uniformly from $1,...,N$ and $N\to\infty$ (this is Dickman distribution). Much later the distribution of other prime divisors was described. This is related to the famous Poisson-Dirichlet distribution. (see also "The Poisson–Dirichlet Distribution and its Relatives Revisited" by Lars Holst).

Leonid Petrov
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When Khare and Wintenberger proved Serre's conjecture, Wintenberger was older than fifty.

Jérôme Poineau
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Connes has initiated whole new areas of mathematics since turning 50: spectral triples, and his novel approach to the Riemann hypothesis, for example.

Abtan Massini
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    But has his approach to RH really yielded progress? My impression from one top expert in analytic number theory is that it ultimately isn't novel (after stripping away the fancy-looking language) and hasn't shed any light on any key issues (after quite some years now). – BCnrd May 23 '10 at 15:36
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    One "idea" of the Connes reformulation is that one can "see" how a dynamical system of primes could prove RH, if one ignores issues like renormalisation and dealing with infinitely many primes rather than finitely many (he proves the S-local analogue of the trace formula). His later programme with Marcolli/Consani has used evermore iffy language and analogues, IMO. On the mathematical end, Meyer had a nice paper on some of the function space constructs, though he doesn't actually try to get RH to appear via his re-working. http://projecteuclid.org/euclid.dmj/1113847338 – Junkie May 24 '10 at 02:35
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    With my admiration for our guru in noncommutative geometry, we must be realistic: his Fields medal-winning discovery of classification of factors of type III and his a bit later single-handed introduction of major characters in noncommutative geometry like the introduction of cyclic homology 30 years ago, while not as perfect as some modern ramifications seem historically far deeper and more striking discoveries than the more synthetic mature (and more collaborative) works at present. – Zoran Skoda May 24 '10 at 14:46
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A recent example (you may or may not think it's a major advance - but it is certainly big news in fundamental game theory): William Press (64) and the legendary Freeman Dyson (89) have shown that iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent (in the paper bearing the same title).

Harun Šiljak
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The Fermat number $F_6$ was shown to have nontrivial factorization, by Landry at the age of 82. And apparently it was Landry's only mathematical publication.

(Source: Ribenboim, Prime number records(the smaller book).)

This is perhaps not a "major mathematical advance" in the sense of Hardy; but is inspiring nonetheless. I have seen a good number of elderly retired people with dreams of solving Fermat's Last Theorem or other such theorems in a simple way, and doggedly keep on trying and without getting disheartened by the lack of recognition for their efforts.

Martin Sleziak
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Anweshi
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This "almost" answers Zoran Škoda's question: Otto Grün (his theorems in group theory are still well known) published his first paper at the age of 46.

Martin Sleziak
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Franz Lemmermeyer
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Andre Weil lay the modern foundation of "theta series" in Acta math. (1964/65) when he was almost 60 years old!

user4245
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Caspar Wessel, a surveyor born in 1745, presented his only math paper "Om Directionens analytiske Betegning" (in Danish) in 1797 at the age of fifty two on complex numbers. His paper was forgotten for almost 100 years until his paper was translated into French in 1878(?). In the meantime Gauss in 1831 and Argand in 1806 re discovered Wessel's idea.

By reading the texts in Complex Numbers you will hardly know the contributions Wessel.

Martin Sleziak
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abel
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    Interesting example, but I think the statement of Hardy is best understood for professional mathematicians (not that I agree with it, btw). – Thierry Zell Feb 15 '11 at 16:49
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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.

Martin Sleziak
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Thomas Sauvaget
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    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.

     – Daniel Asimov May 25 '10 at 01:50
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    By the way, to get the accents in comments you can just copy-paste: Poincaré. – Victor Protsak May 25 '10 at 04:46
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Burnside proved the $p^aq^b$ theorem at age 53.

Alain Valette
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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.

András Bátkai
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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.

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

Martin Sleziak
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Joël
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2

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

Papiro
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2

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.

Martin Sleziak
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2

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 ...

Elemer E Rosinger
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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 03 '10 at 13:05
1

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.

Mohammad Golshani
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1

Grothendieck wrote "Pursuing Stacks", aka his letter to Quillen, at roughly the age of 55.

user1504
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1

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.

Michael Renardy
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