Expected time before a theorem or theory becomes 'easy'

Question

Whenever a (major) new result is proved, it is typically only understood only by a handful of experts, then by people who work in close vicinity, gradually understood by everyone in the general subject area, and then by a broad mathematical audience, and if the result is particularly important, eventually trickle down to mainstream science. This process is a more realistic version of the adage "there are only two kinds of mathematical problems: trivial and impossible. It is impossible until you solve it and then it becomes trivial."

I am asking what is the expected amount of time (in a modern sense) before a major breakthrough in mathematics is 'digested'. For instance, the invention of calculus was no doubt at the time a tour de force, but by now it is understood by almost all professional mathematicians and is taught to a broad audience at a relatively low level. Doron Zeilberger once remarked, on his opinion page, that eventually the proof of Fermat's Last Theorem can be understood by undergraduates.

Of course, I am asking for the average time, as there are great many examples of variances. I believe that the proof that there exist infinitely many bounded gaps between primes will be digested far faster than, say, the proof of Weil's conjectures, due to the relatively simple principles and machinery that goes into the former. Thus a discussion on the variance and the reliance on particular subject matters might be enlightening as well.

Answer 1

This time widely varies, so the "average time" is not well-defined. Once I asked myself: what is the most recent theorem that we teach to undergraduates (in my university). This is the duality theorem in linear programming by J. Nash which is about 50 years old. (Next goes the fast Fourier Transform). Most of the undergraduate math curriculum was essentially discovered in 18s century, with some improvements made in the 19s.

But there are theorems whose proof does not simplify over very long periods. One example is the Uniformization Theorem which is central in the subject of Conformal Mapping. It was discovered by Klein in the late 19s century, and rigorously proved by Poincare and Koebe in the very beginning of the 20s century. The proof was somewhat simplified since then but to give a complete proof is an advanced course remains a challenge. On the other hand, everyone knows this theorem and many people use it.

Classification of simple Lie algebras is another example. It was obtained in 1890s by Cartan and Killing, many simplifications and improvements were made since then, but the result remains difficult and takes long time to explain.

If you want an older example, take a fundamental result of ancient Greek geometry: classification of regular polytopes (including their construction). Of course now, after more than 2000 years passed we have much better understanding of this matter, and we can explain it to undergraduates in 2 or 3 hours, but still I would not say that this is too easy.