Major mathematical advances past age fifty 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.
 A: 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.
A: 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.
A: Roger Apery was 62 when he proved the irrationality of $\zeta(3)$.
A: Andre Weil lay the modern foundation of "theta series" in Acta math. (1964/65) when he was almost 60 years old!
A: 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.
A: 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.
A: Weierstrass approximation theorem was proved by Karl Weierstrass when he was 70 years old
A: 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.
A: 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.
A: 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.
A: 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.
A: 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.
A: 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.
A: 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:

*

*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.


*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: 


*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.

A: Burnside proved the $p^aq^b$ theorem at age 53.
A: 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.
A: Uncle Petros proved Goldbach's conjecture just minutes before his death, when he was more than sixty.
A: 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.
A: 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.
A: 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?
A: 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.
A: Louis de Branges solved the Bieberbach conjecture in 1985 when he was 53.
A: 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)
A: Furtwängler proved the principal ideal theorem when he was almost 60. No small feat given that Artin and Schreier simultaneously were working on it. 
A: Paul Erdős continued to do work in many fields including combinatorics after his 50th birthday. Some of his papers are here
A: 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.
A: 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 ...
A: Fourier (1768 - 1830) presented his work Théorie analytique de la chaleur in 1822 at age 54.
A: 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.
A: 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.
A: 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. 
A: 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
A: 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.
A: Tibor Rado introduced the busy beaver function and proved its noncomputablity at the age of 67. 
A: 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).
A: When Khare and Wintenberger proved Serre's conjecture, Wintenberger was older than fifty.
A: Connes has initiated whole new areas of mathematics since turning 50: spectral triples, and his novel approach to the Riemann hypothesis, for example.
A: 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).
A: 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.
A: Grothendieck wrote "Pursuing Stacks", aka his letter to Quillen, at roughly the age of 55.
A: 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.
