Great contributions to mathematics by older mathematicians [closed]

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!

Solution 1:

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.

Solution 2:

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.