Joseph Howlett in Quanta:
In 1994, an earthquake of a proof shook up the mathematical world. The mathematician Andrew Wiles had finally settled Fermat’s Last Theorem, a central problem in number theory that had remained open for over three centuries. The proof didn’t just enthrall mathematicians — it made the front page of The New York Times(opens a new tab).
But to accomplish it, Wiles (with help from the mathematician Richard Taylor) first had to prove a more subtle intermediate statement — one with implications that extended beyond Fermat’s puzzle.
This intermediate proof involved showing that an important kind of equation called an elliptic curve can always be tied to a completely different mathematical object called a modular form. Wiles and Taylor had essentially unlocked a portal between disparate mathematical realms, revealing that each looks like a distorted mirror image of the other. If mathematicians want to understand something about an elliptic curve, Wiles and Taylor showed, they can move into the world of modular forms, find and study their object’s mirror image, then carry their conclusions back with them.
More here.
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