Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How hard is the proof from the perspective of the non-expert in the field? In the opposite direction, can you give examples of important results, for which several genuinely different proofs were found? Are these proofs all considered equally hard, or some are much easier or, perhaps, even surprisingly easy?

For example, [Carleson's theorem][1] has more than one proof and, although the latest proof is much simplified, it is probably still quite technical. Poincare conjecture has one known proof, not easily accessible to non-experts. Kadison-Singer problem has one known proof, accessible to non-experts with a little effort. [Capset problem][2] has one known proof, fully accessible to non-experts. The last three examples are relatively recent results, and some of them are likely to remain the only known approaches for a long time, while maybe not others. Instead of trying to guess the future, what are interesting examples that are less recent?

  [1]: https://en.wikipedia.org/wiki/Carleson%27s_theorem#History
  [2]: http://arxiv.org/abs/1605.09223