If you haven't heard of Langlands but have heard of Andrew Wiles and his proof of Fermat's Last Theorem, you may be interested to learn that the two are somewha

If you haven't heard of Langlands but have heard of Andrew Wiles and his proof of Fermat's Last Theorem, you may be interested to learn that the two are somewhat closely related. Wiles actually proved a part of the modularity theorem, which had been shown by Frey, Serre, and Ribet to imply FLT in the 80s. The modularity theorem is very much a Langlands-style theorem and could be seen as a more concrete version of many of the ideas and conjectures that form the Langlands program. The conjecture now known as the modularity theorem was formulated as early as the 50s and 60s by Taniyama and Shimura, thus predating the Langlands program, and it was taken seriously once Weil gave conceptual evidence for it (but did not come close to a proof). In fact, the modularity theorem is just a very oddly phrased reciprocity law. General reciprocity laws often look astonishingly nothing like the simple law of quadratic reciprocity, or they require some clever squinting to see the relationship. Modularity gives you for any rational elliptic curve E a modular form which is a simultaneous eigenvector for the Hecke operators (one for each prime number p) and whose eigenvalues give the solution counts of the elliptic curve equation modulo p for various primes p. These eigenvalues are also the coefficients in the Fourier expansion of the modular form.