What Is Math’s Mysterious Langlands Program Really About?
Samuel Velasco/Quanta Magazine
Introduction
The mathematical universe is boundless and heterogeneous, encompassing, over here, truths about numbers and equations; there, the logic of shapes and spaces; and over there, the study of change and probability. Its domains have grown organically over centuries, each centered around its own objects, methods, and questions. That’s why it’s so surprising when direct connections between different areas of math are discovered, like wormholes linking distant galaxies.
The wormholes are known as Langlands correspondences, and the decades-long effort by hundreds of mathematicians to extend and exploit these correspondences is the Langlands program. Because it suggests an underlying unity to the universe of mathematical truths, the Langlands program has been called a “grand unified theory of mathematics.”
Yet even most mathematicians don’t quite know what to make of it. I asked several experts connected to the Langlands program whether they thought the average attendee at the recent International Congress of Mathematicians would have a decent understanding of what it is, or if they would have no idea. “I would think more the latter than the former,” said David Ben-Zvi, a mathematician at the University of Texas, Austin who studies the geometric Langlands correspondence, echoing the typical response. “Everyone will have heard of it, certainly.”
Even among experts, descriptions of the Langlands program sound nothing alike. One said it’s all about “unexpected symmetries.” Another defined it as “bridges between two areas of mathematics.” Someone else said it’s “the best vision we have to understand non-abelian versions of Fourier theory” — a statement I’ll unpack later, because it might be the deepest explanation available so far. References were made to the parable of the blind men and the elephant. The program has so many aspects and corners and consequences that it can be hard to interpret. Mathematicians shy away from interpretation by nature, anyway, since anything they say will be unproven. Another challenge is that though the Langlands program is sweeping and unifying, the mathematical correspondences themselves are excruciatingly specific and esoteric.
Here is my attempt, as someone who has covered math and physics for years, to unpack Langlands from its origins to its current form as a mathematical project with no precedent. I’ll also explore what this set of connections means. It’s not easy, because mathematicians have yet to mine the deepest meaning of the Langlands program — an obscure message that seemingly pertains not only to the mathematical universe, but also to the physical one.
Robert Langlands, a Canadian mathematician now in his 80s who occupies Albert Einstein’s former office at the Institute for Advanced Study in Princeton, New Jersey, launched the program that bears his name in 1967. In a letter to a colleague, he picked up on a connection between far-flung mathematical realms: number theory and harmonic analysis (the study of signals and waves). From there, Langlands conjectured the existence of a family of correspondences between objects, symmetries, and properties.
To understand that original Langlands wormhole, we can start with a collection of numbers called $latex \mathbb{Q}$. Specifically, $latex \mathbb{Q}$ is the set of all rational numbers, meaning those expressible as fractions, such as 1, or $latex \frac{5}{16}$, or $latex – 5.3277$. These form a special kind of set called a number field, because adding, subtracting, multiplying, or dividing rationals (except for dividing by zero) yields a number that is also in the set.
In 1967, Robert Langlands revealed the first surprising correspondence of the sprawling mathematical endeavor that now bears his name.
Dan Komoda/Institute for Advanced Study
But things change when you introduce a polynomial equation, such as $latex x^{2}\kern0.5pt -2=0$. Even though these equations exclusively feature rational numbers in their terms, the solutions — the values of $latex x$ that make them true — usually lie outside $latex \mathbb{Q}$. Our simple equation, for example, has the solutions $latex x = \sqrt{2}$ and $latex x =\kern0.5pt- \sqrt{2}$, which are irrational; their digits begin $latex 1.41421\ldots\ $and keep going forever without repeating.
Mathematicians figured out that they can use these solutions to extend $latex \mathbb{Q}$. We can make a new field, $latex \mathbb{Q}$($latex \sqrt{2}$), by appending to $latex \mathbb{Q}$ the irrational numbers $latex \sqrt{2}$ and $latex -\sqrt{2}$, as well as all the irrationals you can get by arithmetically combining those solutions with other numbers in the field: Any number of the form $latex a + b\sqrt{2}$ (for any rationals $latex a$ and $latex b$), such as $latex 5 + 3\sqrt{2}\ or\ -\frac{\sqrt{2}}{7}$, is in the field. If you picture the possible values of $latex a$ as a horizontal number line and $latex b$ as a vertical number line, the number field $latex \mathbb{Q}$($latex \sqrt{2}$) spans the whole plane.
Crucially, $latex \mathbb{Q}$($latex \sqrt{2}$) has a “symmetry”: a transformation you can do to it that preserves all its elements. Namely, you can replace every $latex \sqrt{2}$ that appears in the field with the equation’s other solution, $latex -\sqrt{2}$, and vice versa. When you do this, the number field remains intact. Every $latex a + b\sqrt{2}$ becomes $latex a\kern0.5pt-\kern0.5ptb\sqrt{2}$, which was included in the field already, sitting on the opposite side of the horizontal axis. So this symmetry transformation is like reflecting the field in a mirror.
The collection of such symmetries is called the Galois group of the equation, after Évariste Galois, a mathematician who studied them in 1832. Galois’ big insight, which came at age 20, just weeks before he got himself killed in a duel, was that these symmetry groups reveal a great deal about equations and their solutions, even for equations that are too hard to solve.
Let’s return to $latex x^{2}\kern0.5pt-\kern0.5pt2 = 0$. Its Galois group consists of two symmetries: the one that swaps $latex \sqrt{2}$ and $latex -\sqrt{2}$ throughout $latex \mathbb{Q}$($latex \sqrt{2}$), and an identity operation that does nothing (like multiplying everything by 1). This Galois group is what is known as “abelian”: You can execute its two symmetries in any order, and the number field $latex \mathbb{Q}$($latex \sqrt{2}$) will end up oriented the same way.
Next we’ll consider $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$, which, though it looks only a tick different than the last equation, is already complicated enough to illustrate what the Langlands program is about (but hopefully no more complicated than that).
It has three solutions, let’s call them $latex {\ x}_{1}$, $latex x_{2}$ , and $latex x_{3}$. These again create an extended number field, and this field has its own Galois group of symmetries — ways to rearrange the solutions — called S3. There are six symmetries: two ways to swap all three solutions, three ways to switch any two of them, and one way to do nothing. It helps (and is mathematically accurate) to picture $latex x_{1}$, $latex x_{2}$, and $latex x_{3}$ as corners of an equilateral triangle. Swapping all three is equivalent to rotating the triangle by 120 degrees clockwise or counterclockwise. Switching two, however, is equivalent to flipping the triangle, thereby exchanging two corners.
Importantly, this combination of symmetries makes the Galois group S3 “non-abelian,” meaning the order in which you transform the field matters. Rotating the triangle 120 degrees and then flipping it puts the three solutions in different positions than if you flip first and then rotate.
The S3 symmetries can be expressed as a set of six little 2 × 2 matrices, or blocks of numbers; this is called a Galois representation. The correspondence Langlands discussed in his 1967 letter is the fact that each Galois representation generates a barcode-like sequence of data that controls the form of a completely different mathematical object, one that differs so utterly in its origins and premise that the connection between them seems almost miraculous. It’s this unexpected link, and other, analogous wormholes, that mathematicians have been exploring ever since.
You have a taste by now of the strange cosmos mathematicians bop around in, thinking ultra-clearly about objects such as $latex \mathbb{Q}$ ($latex \sqrt{2})$ and extending the ideas in them to find new, more complex truths. So far we’ve stayed on the side of the Langlands correspondence concerning numbers and algebra. Now we’ll pass through the wormhole to harmonic analysis, where mathematicians study how signals like lights and sounds can be broken up into component frequencies.
Within the galaxy of harmonic analysis, we land on a special kind of mathematical object called a modular form — an object that, like sound and light waves, can be written as a sum of simple oscillating components. First studied in the 19th century, modular forms are “automorphic,” meaning they have a kind of symmetry where you can shift or transform the input (the value that you plug in to the function) by certain amounts and get the same output. Mathematicians vividly illustrate these functions by assigning colors to different output values and then making a color map of the inputs that yield those outputs. Repetitions in the resulting patterns encode the modular form’s inherent symmetries.
I mentioned that modular forms can be broken up into component parts. The modular form that concerns us here is defined by the following function, built from ever-larger powers of the input variable, $latex q$:
$latex f(q)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}+\ldots$
The Langlands miracle here is that the coefficients in front of the prime-numbered powers of q in this function come from our little 2 × 2 matrices representing the Galois group S3, from our equation $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$. Prime numbers, those divisible only by 1 and themselves, are the atoms of arithmetic; all other whole numbers are built from them. Likewise, the coefficients of the terms with prime exponents shape a modular form. Let’s take a closer look at those coefficients:
$latex f(q)=\ldots+0q^{2}+0q^{3}+\ldots+0q^{5}+\ldots-1q^{7}+\ldots+0q^{11}$
$latex \qquad+\ldots-1q^{13}+\ldots+0q^{17}+\ldots-1q^{19}+\ldots+0q^{23}$
$latex \qquad+\ldots+2q^{31}+\ldots-1q^{37}+\ldots+2q^{43}+\ldots-1q^{61}$
$latex \qquad+\ldots-1q^{67}+\ldots$
They’re all zero, $latex -1$, or 2. (The other coefficients are arithmetic combinations of those, based on the prime factors of the power.) Likewise, each of the matrices in our Galois representation of S3 has a number called a trace, which is the sum of its top left and bottom right elements. And the traces of the reflection, 120-degree rotation, and identity matrices are zero, –1, and 2, respectively.
This visual representation reflects the modular form cited above, which is surprisingly linked to the equation $latex x^{3}\:\!-\:\!2 = 0$.
Mark Belan/Quanta Magazine; Source: David Lowry-Duda for Quanta Magazine
Which trace from our Galois representation clings to each of these terms with prime exponents depends on how our equation, $latex x^{3}\:\!-\:\!2 = 0$, behaves in a number field defined by that particular prime. In each case, the equation is solved in what is known as modular arithmetic. Modular arithmetic works like a clock; instead of using all numbers, it counts up to a certain number — 12 in the case of a clock — and then resets to zero. Counting “modulo 12” (for example) means that numbers such as 13 and 25 are numerically equivalent to 1, and all multiples of 12 would be zero.
The trace that goes in front of $latex q^{31}$, for example, depends on how $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ behaves when the modulus is 31. In that case, the equation has three simple solutions: 4, 7, or 20. (Plugging 4 into $latex x^{3}\kern0.5pt-\kern0.5pt2$, for example, gives 62, which is a multiple of 31, and therefore the same as zero in this limited arithmetic.)
Recall that the Galois symmetries are ways of rearranging an equation’s solutions. To compute the effect of these symmetries in modular arithmetic, you raise each solution to the power of the modulus. In this case, our solutions, raised to the power of 31, don’t change at all; for example, $latex 4^{31} = 4$ (modulo 31). Therefore, the Galois group acts on these solutions as the identity operation, and that’s why $latex q^{31}$ picks up the trace of the identity matrix, 2, as its coefficient.
For comparison, the solutions of $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ modulo 13 are $latex \sqrt[3]{2}$, $latex 3\sqrt[3]{2}$, and $latex 9\sqrt[3]{2}$, which in this case all lie outside the number field defined by this prime. Now, raising each solution to the 13th power transforms it into one of the others. For example, $latex (\sqrt[3]{2})^{13} = \sqrt[3]{2}[(\sqrt[3]{2})^3]^4$ $latex = \sqrt[3]{2}(2)^4$ $latex = 16\sqrt[3]{2}$ $latex = 3\sqrt[3]{2}$ (modulo 13). That’s the 120-degree-rotation element of our Galois group S3, which has a trace of –1. Accordingly, the coefficient of $latex q^{13}$ in the modular form is –1.
The other coefficients of prime-exponent terms that define our modular form are determined by the symmetry properties of the solutions of $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ modulo that prime; it leaves them unchanged if they are all in the number field, or switches the two that are not, or cycles all three solutions, providing the coefficients 2, zero, and –1, respectively. Somehow, Galois symmetries from the faraway galaxy of number theory paint the swirly pattern of this modular form. It’s not a simple connection, but it can’t be a coincidence.
The correspondence I described here goes far, far beyond $latex x^{3}\kern0.5pt-\kern0.5pt2 = 0$ and $latex f(q)$. Langlands conjectured correspondences between all suitable Galois representations and automorphic forms (a generalization of a modular form). Although this sounds esoteric and isolated, it isn’t. Every mathematical object and idea logically relates to many others, so the discovery of a wormhole between distant parts of math, involving such fundamental objects as numbers, equations, groups, functions, and symmetries, has had widespread ramifications. Traversing the wormhole has often yielded proofs or insights about one side with the aid of the other.
In 1994, for example, the wormhole enabled the mathematician Andrew Wiles to prove Fermat’s Last Theorem, one of the most famous and long-standing open problems in math. You’ll recall that the Pythagorean theorem relates the three sides of a right triangle: $latex a^{2} +\kern0.5ptb^{2} = c^{2}$. Fermat guessed that for any whole number $latex n$ larger than 2, there are no three nonzero integers $latex a$, $latex b$, and $latex c$ that satisfy the equation $latex a^{n} + b^{n} = c^{n}$. No one could prove it for 357 years.
What finally worked was proving a Langlands-like correspondence. First the German mathematician Gerhard Frey showed that if Fermat’s Last Theorem is false, then you could use those numbers to write a formula $latex y^{2} = x(x\kern0.5pt-\kern0.5pta^{n})(x + b^{n})\ $defining a so-called elliptic curve. Other researchers then showed that such a curve could not possibly correspond to an infinite sum that’s modular, like our $latex f(q)$ above.
This meant Frey’s curve ran afoul of a Langlands-type correspondence called the Taniyama-Shimura-Weil conjecture, a precursor to Langlands’ 1967 idea that turned out to be a special case of the more general phenomenon he envisioned. The conjecture says that every elliptic curve over the rational numbers corresponds to a modular form. Wiles proved this conjecture true in a large enough class of cases to rule out the existence of Frey’s counterexample, thereby proving Fermat’s Last Theorem.
Analogues of the same wormhole have shown up in other areas of math.
The person to whom Robert Langlands wrote his famous letter, the French mathematician André Weil, had sent a famous letter of his own in 1940, 27 years earlier, to his sister, the philosopher Simone Weil. In it, he had described a vision of a mathematical Rosetta stone with three columns. One column was number theory, another concerned curves over finite fields (a strange kind of geometry that takes place in the world of modular arithmetic), and the third involved the more familiar geometry of smooth surfaces. Weil observed patterns that were appearing in all three areas.
Indeed, each of Weil’s columns is now known to feature its own family of Langlands correspondences: wormholes linking one kind of mathematical object to another very different kind that somehow encodes the same information. In the number theory column, Galois representations connect to automorphic forms in the distant galaxy of harmonic analysis. For curves over finite fields, representations of their own Galois groups are likewise matched with automorphic forms. And for smoothly curving 2D spaces called Riemann surfaces, geometric objects describing symmetries and motions around the surface match more exotic objects of harmonic analysis called sheaves.
Decades of work and many of the field’s major prizes have gone toward proving conjectured Langlands correspondences in various settings. In 2024, for example, a group of mathematicians made headlines with a set of papers — more than 800 pages in total — that proved a major case of the geometric Langlands correspondence. Proving these correspondences also hastened progress in the connected research areas. “We can deduce things in one world using results in the other world,” said Jessica Fintzen, a mathematician at the University of Bonn.
We have to trust our mathematicians when they say there’s no apparent reason for the mysterious correspondences. “It’s the best possible world, but it’s very difficult to actually pinpoint what it means,” Fintzen said.
Ben-Zvi thinks of the Langlands program as a non-abelian version of Fourier analysis, which is one of the most ubiquitous tools in math and physics. As Joseph Fourier figured out in 1807, any signal — a sound wave, say, or a beam of light — can be decomposed into the pure tones or colors that make it up. Mathematically, these pure frequencies are sine waves, which, when shifted by regular intervals, don’t change. Picture sliding a sine wave left or right. If you shift it the right amount, it will appear not to have moved at all. The symmetries of sine waves are abelian; the order in which you shift the wave back and forth doesn’t matter. And in those waves, the coefficients that set the strength of each pure frequency are all simple numbers. Automorphic forms are like more sophisticated sine waves, and they can have non-abelian symmetries. In that case, the coefficients are tied to matrices and Galois representations — the original Langlands correspondence.
So perhaps the Langlands program sharpens a broader mystery of why Fourier decomposition is possible, why it lets us analyze signals, compress images, reconstruct medical scans, and do a million other things. That a complicated object can be broken down into a spectrum of pure components is something of an organizing principle of modern science.
Some of the researchers I spoke to suspect that objects on both sides of the Langlands correspondence might be shadows or facets of some other, still-hidden mathematical object or structure. “There is certainly some expectation that there could be something bigger that explains the kinds of relationships one sees in the Langlands program,” said Ana Caraiani, a mathematician at Imperial College London.
Supporting evidence for that view, and what seems to me to be one of the most important clues about the deep origins of the Langlands correspondence, came from thinking about the physical universe rather than the mathematical one. In the 2000s, the theoretical physicists Anton Kapustin and Edward Witten realized that the geometric Langlands correspondence is a consequence of a “duality” (a situation in which one system has two different physical descriptions) that’s exhibited by certain quantum theories. The simplest instance is the so-called electric-magnetic duality: When there are no charges or currents around, electric and magnetic fields are interchangeable; you could swap them and there would be no way to tell. (The presence of electrically charged particles such as electrons breaks the mirror between them, because no equivalent magnetic charges exist.)
Kapustin and Witten studied a model of space-time (the four-dimensional fabric of the universe) in which two spatial dimensions form a Riemann surface, the kind of 2D surface involved in the geometric Langlands correspondence. The physicists found that switching the electric and magnetic fields in this patch of space-time had the effect of switching between sides of the geometric Langlands correspondence.
That’s hard to parse, but according to Ben-Zvi, it showed that the two sides of geometric Langlands are conceptually close to each other in some way we don’t yet understand. Electric-magnetic duality reflects the fact that both fields are really aspects of a single, underlying quantum “electromagnetic field.” “It’s not two super-exotic worlds like Galois groups and automorphic forms, which sound totally unrelated,” he said, but rather a pair of intertwined quantum fields. Similarly, the various Langlands correspondences may eventually be understood as dual views of a single object or system.
Edward Frenkel, a mathematician at the University of California, Berkeley who has worked on the geometric Langlands program for decades, suspects so. “The real reason in my view is that there are things below the surface that have not yet been discovered,” he said.
On our video call, he held up a hot pink coffee cup, and we considered its shadows. “The projection onto the table will be a disk,” he said. “A projection on the wall will be a rectangle. And then you start marking and say, ‘Oh, there is a point here that connects to a point here.’ To you it appears surprising that points in one projection or shadow correspond to points in the other. But if you find the real source of this, if you start seeing the source and not just the projections, that would give you a much more convincing explanation of this duality or correspondence.
“Consciously or unconsciously,” Frenkel added, “people are excited about these connections precisely because they point to some deeper structures in mathematics that we have not found. Eventually we hope to find them, by finding more and more information which hopefully will lead us to the true explanation, the true reason.”
