After Yitang Zhang stunned the mathematics world by establishing the first finite bound on gaps between prime numbers, it got me thinking about the following question:

$\underline{\text{Question}}:$ What are other examples of proofs provided by younger, less accomplished mathematicians that the experts of the time could not solve or did not attempt to solve?

For example, in 1979 the American mathematician Thomas Wolff created a new proof of the Corona problem whilst still a graduate student at Berkeley. He solved the equations in a nonanalytic way and then modified the solution to make it analytic bounded (which leads to the equation $\overline{\partial}u=f$).


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    $\begingroup$ Yitang Zhang proved his result on bounded prime gaps at the age of 58 (or thereabouts). 'Less accomplished', maybe, but certainly not 'younger'. $\endgroup$ – John Gowers May 4 '15 at 10:16
  • $\begingroup$ I know that, but it got me thinking about mathematicians that are a lot younger that stunned the mathematics world at an early age $\endgroup$ – user230715 May 4 '15 at 10:18
  • $\begingroup$ I want to say Fermat's last theorem, because while Andrew Wiles was an established mathematician, many of the brightest brains for the previous three and a half centuries couldn't prove it. On the other hand, Wiles' theorem was just the last link in a chain of results with the end conclusion that Fermat's last theorem was true, so it's not all his achievement. $\endgroup$ – Arthur May 4 '15 at 10:18
  • $\begingroup$ That is very true. One example (if I recall correctly) is that Ken Ribet of Berkeley settled the epsilon conjecture which was necessary in proving Fermat's last theorem. $\endgroup$ – user230715 May 4 '15 at 10:19

Kurt Heegner showed the Stark-Heegner theorem. At that time, he wasn't connected to any university, in fact, no one looked to his proof until Stark showed the same result. He wasn't young, but Yitang Zhang wasn't either.