In January 1967, Robert Langlands sat down and wrote seventeen pages of mathematics by hand, then mailed them to André Weil. He opened with an apology and a proposal: "If you are willing to read it as pure speculation I would appreciate that; if not — and that is more likely — please treat it as correspondence in the mathematical waste basket." He was thirty years old. Weil was sixty, one of the most formidable mathematicians alive, and the letter contained what Langlands himself could only describe as speculation, hedged and tentative. He was not sure it was worth Weil's time.
It was not discarded. What Langlands had written down, apologetically, in seventeen handwritten pages, was the germ of the grandest unifying vision in the history of mathematics — a conjectured correspondence between two worlds so different that connecting them seemed almost presumptuous. On one side: Galois representations, objects from number theory and algebra, encoding the symmetries of solutions to polynomial equations. On the other: automorphic forms, objects from analysis and harmonic analysis, functions on certain symmetric spaces with enormously rich structure. These two worlds had developed almost entirely separately. Langlands was proposing that they were, in some deep sense, the same.
The audacity of this is easy to miss if you don't feel the gap between the two sides. Galois representations come from asking about symmetries of number fields — the hidden permutations that shuffle roots of polynomial equations while preserving all algebraic relationships. They are creatures of pure algebra and arithmetic. Automorphic forms are creatures of analysis: they are functions satisfying complicated transformation laws under the action of arithmetic groups, generalizations of the way sin and cos transform under translation. The one side lives in the discrete world of integers and field extensions. The other lives in the continuous world of functions, integrals, and harmonic decompositions. The connection Langlands proposed is that both sides produce the same mysterious objects — L-functions — and that this coincidence of L-functions is not a coincidence at all.
The clearest window into what this means is the GL(2) case, where the correspondence becomes almost tangible. A modular form is a holomorphic function f(z) on the upper half-plane satisfying a precise symmetry condition under a certain group of fractional linear transformations. Such functions have Fourier-like expansions: f(z) = a₀ + a₁q + a₂q² + a₃q³ + ..., where q = e²ᵖⁱᶻ. The coefficients aₙ are, at first glance, just numbers — the specific values that happen to appear in the expansion of this particular function. But for a normalized Hecke eigenform, the coefficients at prime indices — a₂, a₃, a₅, a₇, ... — turn out to carry precise arithmetic information. Specifically, for each prime p, the coefficient aₚ is exactly the trace of Frobenius in the corresponding two-dimensional Galois representation.
The Frobenius element at a prime p is a canonical symmetry in the Galois group — a specific permutation determined by how p behaves in the number field extension. Its trace is a single number summarizing its action. And that number, computed entirely on the Galois side through pure algebra, equals aₚ, computed entirely on the automorphic side by expanding an analytic function as a power series. The same number, derived by two utterly different methods. This is not a numerical coincidence to be explained away. It is the Langlands correspondence, made concrete enough to check by hand.
Fermat's Last Theorem is the most famous consequence. The proof, completed by Andrew Wiles in 1994, runs through the GL(2) Langlands correspondence in a specific and beautiful way. Gerhard Frey had observed in 1986 that if Fermat's equation aⁿ + bⁿ = cⁿ had a nontrivial solution, you could construct from it an elliptic curve with bizarre properties — properties that seemed to violate everything known about the arithmetic of elliptic curves. Ken Ribet proved in 1987 that this curve, if it existed, could not correspond to any modular form. And Wiles proved that every semistable elliptic curve over the rationals does correspond to a modular form — that every such curve IS modular, in the precise Langlands sense. The chain closes: a solution to Fermat's equation would produce a non-modular elliptic curve; non-modular semistable elliptic curves don't exist; therefore neither does the solution. Fermat's Last Theorem is a corollary of the modularity of elliptic curves, which is a special case of the GL(2) Langlands conjecture over Q.
What makes the program so ambitious — and so difficult — is that this is just one small piece. The full Langlands program covers all reductive algebraic groups, not just GL(2). It covers all number fields, not just the rationals. And it introduces a further layer of structure that is beautiful in its own right: the L-group, or Langlands dual group, constructed by exchanging roots and coroots of a reductive group G. For GL(n), the dual group is again GL(n,ℂ). For the symplectic group Sp(2n), the dual is the odd orthogonal group SO(2n+1,ℂ). The exceptional group E₈ is self-dual. This duality was already lurking in Langlands's 1967 letter — Langlands himself later described the L-group appearing there "like Gargantua, surprisingly mature." The principle of functoriality then says: a map between L-groups should produce a transfer of automorphic representations from one group to another. This is a sweeping organizing principle that connects different parts of the program to each other.
Then, in 2006, something unexpected happened. Edward Kapustin and Edward Witten, working in theoretical physics, showed that the geometric version of the Langlands correspondence is the same thing as S-duality in a twisted version of N=4 super Yang-Mills theory. S-duality is an electric-magnetic duality: it swaps the coupling constant g with 1/g, exchanging strong coupling with weak coupling. And it turns out that the Langlands dual group G^L had been discovered independently by physicists studying magnetic monopoles — the dual group describes the monopole sector of the gauge theory. This is not an analogy. Kapustin and Witten were saying that geometric Langlands is S-duality: these are the same mathematical structure, discovered by two communities with no overlap in methods or motivation. The convergence is structural identity, not metaphor.
The geometric Langlands conjecture — the version reformulated over complex curves rather than number fields — was finally proved in 2024 by Dennis Gaitsgory, Sam Raskin, and seven collaborators, across five papers totaling more than 800 pages. The achievement required developing entirely new frameworks in derived algebraic geometry and the theory of infinity-categories. It is a landmark. But it is worth being clear about what it does and doesn't say. The geometric column of the Langlands Rosetta Stone — curves over the complex numbers — is now largely settled. The arithmetic column — number fields, the thing Langlands was actually writing about in 1967, the world that contains Fermat's Last Theorem — remains almost entirely open beyond the GL(2) case. David Ben-Zvi said the geometric proof had "encircled the problem from every direction," but the original problem is still out there, waiting.
This asymmetry is worth sitting with. Mathematicians have now proved an enormous generalization of the Langlands program, in a setting (geometry over ℂ) that is in many ways more tractable than the arithmetic setting Langlands originally had in mind. The 800-page proof uses tools that didn't exist in 1967 and barely existed in 2010. And yet the original conjecture — the one Langlands wrote down in seventeen handwritten pages and asked Weil to throw away — is still a conjecture.
What I find most interesting about the program is the question of what it means for two mathematical worlds to be "the same." The classical Langlands correspondence is about bijections: this Galois representation corresponds to that automorphic form. Modern formulations push toward something stronger — equivalences of categories, where entire structural frameworks on one side are identified with structural frameworks on the other. The question is no longer "which object corresponds to which?" but "are these two mathematical universes equivalent as categories?" This is categorification, the move from counting things to understanding the spaces of relationships between things. It is the same philosophical step that took group theory from a list of symmetries to a category of representations — not just naming the symmetries but encoding how they compose and transform into each other.
What Langlands glimpsed in 1967, in seventeen pages he thought might be worthless, was that arithmetic and analysis are not separate domains that happen to overlap. They are, in some deep and still not fully understood sense, different descriptions of the same underlying structure. The proof of that in the geometric setting is done. The proof in the arithmetic setting — the original one, the one that would fully justify the audacity of the letter — is still waiting to be written.