Great contributions to mathematics by older mathematicians It is often said that mathematicians hit their prime in their twenties, and some even say that no great mathematics is created after that age, or that older mathematicians have their best days behind them.
I don't think this is true. Gauss discovered his Theorema Egregium, a central result in differential geometry, in his fifties. Andrew Wiles proved Fermat's Last Theorem in his thirties. 

Post many examples of great mathematics created over the age of
  30, the older the better. Bonus points for mathematicians whose
  best work happened over the age of 30. I will define great mathematics to be something of great significance, such as proving a
  deep theorem, developing far-reaching general theory, or anything else
  of great impact to mathematics.
Addendum: Please also include a brief explanation of why the mathematical result posted is significant.

(Many say that 30 isn't that old, but I'm casting the net wide to get more examples. Age-30 examples would also help to debunk the "peak at twenties" myth. I do ask for examples to be as old as possible, so the lower bound isn't that important.)
I know that mathematicians can produce a lot of work throughout their lives - Euler is a great example - but here I want to focus on mathematics of great significance or impact.
Help me break this misconception!
 A: I think not only proper examples would help you to break this misconception but also the time itself. Everything is changing. Some fifty years ago there were no computers in people's life and people were dying earlier. Now it is the opposite. The education system also has changed with the advances in mathematics. The things that are learned nowadays are getting more and more advanced and it also requires alot of time to come to the very deep of the problem. The proof of Fermat's last theorem is a supporter of what is said here. His corrected version of the proof was published on 1995 and he was born in 1953. There is a documentary movie which shows how much effort he put into this problem but in this context more importantly when he decided to prove the theorem..
A: Kolmogorov–Arnold–Moser theory : 
http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem - Kolmogorov was 50 at that time.
EDIT: I decided to edit my answer and add P.S.Novikov here (I only mentioned him in the comment earlier). He was a consistent achiever throughout all his life, but here I want to mention his (negative) solution of word problem in group theory. He proved the result at the age of 54.
A: The Weierstrass approximation theorem was published in 1885, when Karl Weierstrass was 70. As T.W. Körner writes in his Fourier analysis book (pg. 294):

Fejér discovered his theorem at the age of 19, Weierstrass published this theorem at the age of 70. With time the reader may come to appreciate why many mathematicians regard the second circumstance as even more romantic and heart warming than the first.

A: I think that Leonard Euler "prime" years were when he was getting older until he died at 76. Not only did age not stop him but neither did blindness. 
Serre is nearly 90 years old, and he still does mathematics. Of course, not to the same degree of magnitude as he used to do because of much poorer health, but he is still a member of the living mathematical community and we greatly benefit from him. 
A: Victor Volterra 
Born in 1860, he began to develop the theory of dislocations in crystals that was later to become important in the understanding of the behaviour of ductile materials... when he was 45.
Alfred Lotka
Born in 1880, one of his earliest publications, proposed a solution to Ronald Ross's second malaria model (Lotka was 32). He published a thorough five-part analysis and extension of both Ross's malaria models... when he was 43.
Augustin-Louis Cauchy
Born in 1789, he presented a dozen papers on celestial mechanics to the Academy... when he was 51.
Phyllis Nicolson
Wikipedia has little info about this lady. But it is remarkable she started research fellowship after she was 30.
Grace Hopper
Not a mathematician, but belongs to a related field (computer science). Born in 1906, she achieved the inaugural Computer Sciences Man of the Year award from the Data Processing Management Association when she was 63. She is also the mother of Cobol programming language and the term "bug" for an error in a computer system/software.
A: Weierstrass proved his famous Approximation Theorem at age 70.
Tibor Rado introduced "Busy Beaver" functions at age 67.
Freeman Dyson recently published an important contribution to the iterated prisoner's dilemma at age 89.
A: This 'misconception' is not proven or disproven by quoting specific counter examples, but by examining a fair number of mathematical discoveries and determining a trend.
17 equations that changed the world
I will be going by this list based on this book by Ian Stewart of the 17 equations that changed the world, I will be taking the earliest possible date that seems to make sense, because to disprove something that seems to be the most sensible approach. Additionally some of these equations aren't purely mathematical, but that's the nature of mathematics that per the OP is most influential, had the greatest impact.
Please understand, I am not judging whether these discoveries are rightly attributed, I simply followed the list!


*

*The logarithm and its identities - John Napier aged 64 published his work Mirifici Logarithmorum Canonis Descriptio (can't seem to find anything about when he actually first formulated the basis)

*The fundamental theorem of calculus - Newton aged 23 and Leibniz aged 28

*Newton's universal law of gravitation - Newton aged 44

*The origin of complex numbers - Hammilton aged 38

*Euler's formula for polyhedra - Euler aged 43

*The normal distribution - Quetelet aged 39 published his work Sur l'homme et le développement de ses facultés, ou Essai de physique sociale

*The wave equation - Bournoulli aged 38 published Hydrodynamica and Jean D'Alembert aged 29

*The Fourier transform - Fourier aged 43

*The Navier-Stokes equations - Navier aged 37

*Maxwell's equations - Maxwell aged 30

*Second law of thermodynamics - Too messy attribution

*Einstein's theory of relativity - Special Relativity aged 26

*The Schrödinger equation - Schrodinger aged 40

*Shannon's information theory - Shannon aged around 31

*The logistic model for population growth - Messy attribution and discovery

*The Black–Scholes model - Black aged 35 and Scholes aged 32
Conclusion
Historically speaking the greatest mathematical discoveries were made between ages 25 and 45 (average of 35-37). As life expectancy post child years were similar to ours we do not need to normalize for that. It is however possible that in more recent years ages have gone down, leading to the stated believe, however I will not further examine this as it is hard to examine this objectively within the scope of the stated question. Either way, an age of 35 on average still can be considered fairly young, although it is clearly older than the questioned twenties the OP was skeptical about.
Sources
I grabbed all the various years from Wikipedia and did the calculations myself.
A: Louis de Branges proved Bieberbach's Conjecture at age 53.
LINK wikipedia

(I found the picture when researching his birthdate)
A: Grigori Perelman was 36 when he released his proof of Thurston's geometrization conjecture.

A: I don't know about Mathematics in particular, but one observation that may provide an explanation is that at least in some fields of research today, it seems the longer you stay in academia, the more the focus of your work shifts from actual research to administration and "politics".
As a PhD student and maybe post-doc, you might just be peaking at the sweet spot between having already acquired enough knowledge/insight to make meaningful contributions and not yet being drawn into the swamp of academic reality (lecturing, writing grant proposals, networking, etc.). For a lot of older researchers, at least in my experience, their research contribution mainly consists in supervising new generations of young researchers -- which is no small feat, by all means, but might explain why their personal research output declines.
So perhaps the reason is not so much the degradation of intellectual power, but simply given by the pragmatics of not having as much time to dedicate to research any more as when you're young.
Take this answer with a grain of salt, though, as I'm broadly generalizing. A lot of counter-examples have already been given in this thread by yourself and others.
A: George Marsaglia was still active and making significant contributions to the art of pseudo-random number generators with long periods at the age of 86, just shortly before he died. 
Maybe these were not "great" contributions since I could actually understand them :-)
