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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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Such questions are usually counted "community wiki". –  Wadim Zudilin May 23 '10 at 7:01
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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 8:10
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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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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
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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 2 '10 at 0:09
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37 Answers

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

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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" 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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"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 2:43
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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 3:36
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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 3:49
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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.

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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.

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

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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.

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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 7:47
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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.

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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.

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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
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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.

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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)

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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.

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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 8:51
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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?

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+1 for the last line especially. –  Mark Meckes May 23 '10 at 18:24
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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.

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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
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According to wiki, Mihailescu got his PhD at the age of 42; and then proved Catalan's conjecture (http://en.wikipedia.org/wiki/Catalan%27s_conjecture) in 2002, age 47, so almost 50.

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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.

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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.

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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.

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For the lazy, 2013-1955=58. –  Federico Poloni Oct 27 '13 at 12:44
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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.

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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. –  Andrew L Jul 19 '10 at 6:04
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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 8:59
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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.

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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

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

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

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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.

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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. projecteuclid.org/euclid.dmj/1113847338 –  Junkie May 24 '10 at 2:35
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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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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

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

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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.

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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).

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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).

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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!

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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.

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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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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 1:50
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