Question

Major mathematical advances past age fifty

Answer 1

I think this could be similar to what happens with music composers/bands.

Young people will have a few ideas and their career will be built on their new ideas. The first few records from bands are often their best and most innovative works and as they get older they just don't have any more significant ideas.

Now on the other hand in more technical musical composition composers do their best work in their later years of life; however (and this is important) those compositions are *almost always* mature derivatives of their first big ideas.

I'd expect this to be similar in most fields including mathematics. Major new conceptual discoveries are probably concentrated around the early years of mature skill levels; however I'd expect the most technically mature (but probably not outright breakthroughs) to come from much more experienced practitioners.

The age thing could also be a bit of a spectre in the data. Original ideas could come from original ways of viewing the world and those "breakthroughs young men have" could just as easily have come in their 40s if they had learned mathematics later in life. Obviously that's total speculation but its worth considering I'd think.

Answer 2

Just my own humble opinion, but I would argue the opposite: mathematics is so technical and - depending on what field one goes into - the amount of time that it takes to develop mastery sufficient to create new mathematics can be up to several years. In my experience the theorems that are the most deep are the ones that take more than technical expertise. They require insight into the heart of the mathematics and a view of the overall structure of *many* results that fall under the umbrella of the theory. This kind of thinking is more than just mastery - it's keen intuition, near encyclopedic knowledge of the current field, and technical fluency rolled into one. Certainly, I personally know of people making new and relevant mathematics well into retirement....

But one can also come up with plenty of counter examples of this hypothesis. Eugene Wigner (b. 1902) proved his famous "Wigner's Surmise" in 1952. Minoru Tomita's (b. 1924) manuscript on Modular theory (1967) was reinterpreted by Masamichi Takesaki (b. 1933) and is of fundamental importance to the theory of type III factors.  Kyoshi Ito (b. 1915) published his work on stochastic integration in 1944 which forms the foundation of the entire theory of stochastic SDE. Roger Penrose is in his early 90's and is still working. Florence Nightingale (1820) used novel data visualization techniques in the late 1850's (from what I can tell; the Wikipedia account of her work in statistics is vague). Julia Robinson (1919) and collaborators showed that no algorithm exists that can tell whether a Diophantine equation has any solutions in integers in 1970. ... The list goes on.

Answer 3

Simon Donaldson proved the Yau--Tian--Donaldson conjecture at 52 in 2012.

Some of Atiyah's best work in the 80's came after 50. He wasn't proving very technical things like back when he proved the index theorem but made penetrating contributions to gauge theory and TQFTs. I think that is the way of it, especially in the deeper parts of maths, that age brings the ability to make very important *general* contributions, but the stamina to prove very good *specific* theorems is a bit more of a young person's game.

Answer 4

Erdos didn't slow down at 50.

Answer 5

As far as I know the threshold for cognitive decline is 60, so seem likely.

Answer 6

Do you think this is unique to mathematics, or is generally true for great scientists and inventors that most of the time (always with some notable exceptions), their great work is done before 50?

Answer 7

Impressive list.

Answer 8

George Green was pretty old when he made his contributions.