Comment by seanhunter

Langlands is a massive programme but originated with Langlands conjecturing in 1967 about some deep connections between two seemingly unconnected areas of mathematics (number theory and harmonic analysis). In particular, he started with what was then called Taniyama-Shimura-Weil conjecture which concerns the number of integer solutions to particular types of equations of elliptic curves following (for no immediately obvious reason) a harmonic series. Wiles’ proof of FLT proved a special case of TSW and involved techniques which some of his students then used to prove the TSW conjecture in full generality, so is now called the “modularity theorem”.

In fact the Langlands programme began with Langlands writing his ideas in a letter which he wrote to Andre Weil because of his association with the TSW conjecture.

I enjoyed this explanation https://youtu.be/4dyytPboqvE?is=VFFjLHSk7vgosN54 by Ed Frenkel.