One morning in late May, in an auditorium at the University of Oslo, Robert Langlands, one of the most influential mathematicians in the world, delivered a lecture to a crowd of distinguished colleagues. They had gathered to mark the eighty-one-year-old’s receipt of the Abel Prize—what many consider the equivalent of the Nobel Prize for mathematics.
Sharpie in hand, Langlands paced back and forth, looking mostly toward the floor, and began speaking about the field of research that bears his name. “The question is,” the Canadian mathematician told the assembled researchers, professors, and students, “in its present form, do all aspects of it enjoy my approval? And the answer is no.” Some in the crowd chuckled. “If someone could just remove my name from that part and give it another name, I’d appreciate it very much.”
For the relatively sedate world of academia, this was the equivalent of an acclaimed director accepting an Oscar for best movie only to take the stage and lambaste the movie’s sequel, directed by someone else. Near the front of the auditorium sat a man named Edward Frenkel—a Russian-born, California-based mathematician who has spent much of his career expanding on and popularizing Langlands’s work.
In the 1960s, Langlands discovered a relationship between two fields generally considered to be separate: number theory (the study of integers) and harmonic analysis (the study of continuous phenomena such as waves made by the oscillation of a guitar string). His work—which established a strong and unexpected bridge between the realm of whole numbers and the realm of continuous functions—created the possibility for a new way of thinking about connections in mathematics altogether.
Langlands’s observations are so significant they have inspired hundreds of scholars, many of whom have spent their careers refining and proving them, and launched whole new areas of research. One of these new fields—in which Frenkel is a leading scholar—expands upon Langlands’s original observations by extending them to geometry, which, in turn, introduces the possibility of applications in other disciplines. Together, the various lines of investigation (the so-called classic Langlands Program and the expanded geometric and physical Langlands programs) comprise one of the most ambitious mathematical research projects in existence—one over which Langlands himself has no control.
The gap between Langlands and Frenkel, who teaches at the University of California, Berkeley, is shaped by their different technical approaches to pure mathematics and also by their motivations: some of Frenkel’s work deals with the potential “usefulness” of mathematics, which Langlands isn’t so concerned with. This is most apparent in applications of the geometric (but not the classic) Langlands Program to theoretical physics—which, with its captivating notions of space-time exploration and quest to understand the building blocks of the universe, can capture the imagination more than esoteric mathematical initiatives do.
The two mathematicians have also come to symbolize a greater chasm in the academic realm: the difference between fiddling with abstract concepts for the pure enjoyment of the exercise and doing it to solve concrete problems in the real world . As universities shift away from their traditional role as centres for pure research and increasingly favour more pragmatic and applied disciplines (less philosophy, more engineering), the question of what purpose mathematics serves—or whether it needs to serve one—has very real consequences for its practitioners. Against this backdrop, the various interpretations of the Langlands programs have become a case study for debates about the justification of abstract science.
In 2013, Canada’s then minister of science and technology, Gary Goodyear, told the CBC that his department was asking scientists to “appreciate the business side of science. . . . Knowledge that is not taken off the shelf and put into our factories is actually of no value.” And so, says Michael Harris, an American number theorist, pure mathematicians sometimes participate in a collective myth to secure their survival: they speak about their research as imminently applicable. The pressure many mathematicians feel to make the results of their research either as romantic as black holes or as practical as stock brokerage has begun to rid that research of its freedom.
Robert langlands grew up in the Lower Mainland of British Columbia in the 1930s and ’40s. Bored in class, he thought of dropping out for a time. After a high-school teacher encouraged him to continue his studies, Langlands attended the University of British Columbia, then Yale. Now a professor emeritus at the Institute for Advanced Study—he occupies one of the former offices of Albert Einstein—Langlands spends his time in Princeton and sometimes Montreal. His speech is slow and calculated, and he has the kind of harsh, deadpan humour that is particular to certain self-aware intellectuals; he will often suggest to an interviewer what they really meant to ask or should have asked instead. (From Langlands’s reply at the end of one printed interview with a UBC graduate student: “I have already exhausted both myself and you by answering your very few questions at great length. You will be grateful that you asked no more. So am I.”)
Frenkel, whose looks inspired one newspaper to call him “the world’s sexiest spokesman for mathematics,” is a much more natural emissary to the outside world: he has a soft voice that inspires comfort, and he tends to tell his audiences that they have good “energy,” that mathematicians are a “family,” that he speaks “from the heart.” In 2013, Frenkel published a bestselling general-interest book about mathematics called Love & Math: The Heart of Hidden Reality. The book tells the story of Frenkel’s life as a young Jewish man in Moscow, and it explains, in (relatively) simple terms, the Langlands Program and Frenkel’s own work on it. In the book, Frenkel uses familiar metaphors to coax his readers into the world of math: finding irrational numbers is like adding cubes of sugar to a cup of tea; quantum duality is like the relationship between potatoes and onions in his mother’s borscht recipe; the Langlands Program is like a giant jigsaw puzzle.
Frenkel has written columns for the New York Times and has been a guest on the Colbert Report. He costarred in his own semierotic movie about the beauty of mathematics, and he is prone to telling reporters things like “I will show you the ecstasy of mathematics.” So when a journalist writes about the Langlands Program, they generally don’t interview Langlands; they interview Frenkel.
“Mathematicians have miserably failed, as a profession, in terms of communicating with the general audience,” says Frenkel. “But also within mathematics we fail completely in communicating with each other. . . . In my mind, it’s our obligation to share.” Langlands is not much inclined to this kind of sales effort.
If mathematics is a “universal” language, as many popular science books claim (2+2=4, whether it’s written in English or Russian), then it is also one of the least accessible—there is an overwhelming number of dialects, rules, and exceptions. Different fields of study within mathematics use different approaches and vocabulary, and it’s rare for a researcher in one area to fully absorb work in another. Langlands is considered an exceptional talent because of his proficiency in several disciplines. The work he developed in 1967 is so sophisticated that, still today, only a few people can comprehend it. (Langlands was up for the Abel Prize at least three times before it was actually awarded to him; the scholars who nominated him think it was in part because the committee had a hard time understanding his work.)
Historically, pure mathematicians have almost always embraced the futility of their most abstract inquiries. They would often describe their research as the untarnished pursuit of beauty or universal truth. The number theorist G. H. Hardy wrote in 1940 that “real mathematics . . . must be justified as arts if it can be justified at all.”
That kind of engagement with the philosophical underpinnings of mathematics is less common now, and today even some pure mathematicians will talk readily about practical extensions of their work. Those who are interested in computer science or theoretical physics, for instance, also argue that while the abstract might not be immediately applicable, it will almost always develop into something useful—in the same way that non-Euclidean geometry eventually helped Einstein develop his theory of general relativity and number theory is now instrumental to password security.
For Langlands, scientific pursuits such as geology or biology help us “deal with the world and the universe as they are”; the Langlands Program, on the other hand, is “only important insofar as the math as such is important.”
When he started doing research in the ’60s, Langlands was simply investigating potential extensions of class field theory, a branch of pure mathematics. The most obvious applications of his work are in other areas of often equally esoteric mathematics. Andrew Wiles, the number theorist who, in 1995, published a proof of Fermat’s Last Theorem—then in the Guinness Book of World Records as the “most difficult mathematical problem”—solved it with help from the Langlands Program. Most mathematicians also agree that the Langlands Program could help find a proof for the Riemann Hypothesis, probably the most famous unsolved mathematical problem (about the distribution of prime numbers). These problems are just as abstract as Langlands’s own work, however, which means his research program as it was originally conceived has little relevance to everyday life. As Langlands bluntly puts it, his work “could disappear without any trace, and the world would go on as it was before.” (The most basic—yet still painfully inaccessible—explanation of Langlands’s calculations is that he related Galois groups, which are number field extensions, to automorphic forms and representation theory of algebraic groups.)
In the 1980s, mathematicians around the world, including Frenkel, started working on the geometric Langlands Program: a related but parallel field of study that explores some applications of Langlands’s findings. Geometry and number theory had long been considered separate disciplines: geometry comes mostly from the Greek tradition, while algebra originated largely in the Middle East during the golden age of Islam. Even after René Descartes famously merged the two subjects in his 1637 La Géométrie, they rarely overlapped: algebra was considered a pure subject, while geometry, because of its various practical uses, was often treated as a kind of “mixed” pursuit—what we would now call applied mathematics. A variety of connections between geometry and number theory have developed since, but few are as extensive as what the geometric Langlands Program is hoping to construct.
The expansion of the geometric program also paralleled many developments in the world of theoretical physics, which sets out to explain the most fundamental particles and forces in nature; in part because of this progress, physicists and mathematicians have been collaborating more in the past decade than ever before. In the 1970s, physicists developed a theoretical framework that attempts to explain all of the known forces of the universe at once. Called the Standard Model, it has been incredibly successful in predicting phenomena (including the Higgs boson, the famous “God particle,” which could be responsible for all the mass of the universe) that have since been confirmed to exist.
What the model does not do, however, is account for the forces of gravity or dark matter (a mysterious substance—if it even is a substance—that is thought to make up a large part of the universe). In efforts to address this and other issues, researchers have expanded on the Standard Model by inventing the principle of supersymmetry, which predicts a specific relationship between particles. But despite numerous costly experiments, that relationship has never been shown to exist. “This is a little bit of a crisis,” one mathematician said recently. Theoretical physicists tackling this problem today work with a couple of different types of theories (“supersymmetric gauge theories” and “supersymmetric quantum field theories”) that attempt to describe the connections between particles in mathematical terms. And that, says University of Toronto mathematician Joel Kamnitzer, is where the geometric Langlands Program can help (though, he concedes, “I can’t say I completely understand why”).
Though it can be as abstract as pure mathematics, theoretical physics has found a much more established place in the public consciousness, in part because of its enticing visual elements and its offer of adventure. Science fiction has exposed even the most math averse to ideas about black holes, wormholes, and nth dimensions—what could be more compelling than the prospect of travelling the universe and coming back only one day older?—and enormous amounts of money are poured into investigating phenomena such as dark matter. We seem to be much more willing to face the abstract when it can fuel our imagination of the fantastical. This may explain why the outside world has been so much more interested in Langlands’s work since Frenkel came along.
Theoretical physics starts with conjecture that can lead to experimentation and refinement; mathematics, on the other hand, is based on axioms and proofs, and rejects any kind of speculation. This means that “physicists and mathematicians have settled on a fine division of labour in which the former complain about the finickiness of the latter, and the latter complain about the sloppiness of the former,” theoretical physicist Sabine Hossenfelder wrote in her recent book Lost in Math. Most mathematicians working on the Langlands Program have no idea what theoretical physicists are working on, and the extent to which the Langlands Program could be useful to physics is still unknown.
Connecting the geometric theory to physics can, nonetheless, make it more enticing. Popular coverage of the geometric and physical Langlands programs makes it sound like there’s a wide-sweeping collaboration between scientists, where the edges of abstract mathematics and physics have finally conjoined—in an ideal platonic convergence of truths—such that the secrets of the universe will be uncovered in mathematical terms. Still, a mathematician who claims that a project like the Langlands Program has “any physical significance,” Langlands says, is inevitably “going to disappoint people.”
Langlands spent every morning, seven days a week, for five years working on the paper he delivered in Oslo. It is written entirely in Russian and dedicated in large part to reformulating the geometric program championed by Frenkel. This new paper is an attempt to shift the field toward a more traditional approach: it proposes a new mathematical basis for the geometric theory that relates more closely to Langlands’s own conjectures by using similar tools to the ones he used in the ’60s—in the process, restoring his work back to its original arithmetic purity.
Before giving his talk, Langlands sent portions of the paper to various mathematicians familiar with his research. Several months later, he hasn’t heard back from anyone who’s been able to understand it in its entirety.
“If he wants more people to pay attention to it,” says James Arthur, a mathematician at the University of Toronto, “why on earth would he write it in Russian?” Some people have theorized that it was targeted at Frenkel, though Langlands insists that’s not the case—he’s always wanted to write something in Russian, he says, and his retirement had finally furnished him with the opportunity to do so.
If Langlands’s new work is solid, understood by enough people, and adapted into a generalization, it could be “revolutionary,” says Julia Gordon, a mathematician working at UBC. By suggesting ideas that are related to the original Langlands Program, Langlands “is trying to bring us back down to earth,” she says. His recent formulations likely won’t replace the current geometric theory—that one is “here to stay,” Arthur says—but it could become the basis for a new line of study with different potential discoveries.
Frenkel received the complete paper in April, and in June, he emailed Langlands with a few clarifying questions. Langlands didn’t answer them; he just told Frenkel to read the paper again and pay particular attention to a line on page fifty-nine. “I was tired,” Langlands says. “[Frenkel] didn’t seem to be taking account of the essential step, the turning point.”
Months later, Frenkel is “a little bit perplexed,” he says. “Langlands is my hero—I don’t want to criticize him.” But he’s still disappointed by Langlands’s characterization of the geometric and physical programs as speculative and unconvincing—possibly the harshest insult one can make in the realm of mathematical research. Frenkel’s reading of the Russian paper can’t be separated from what was said that day in Oslo; he retorts that Langlands’s latest work is “way less rigorous than what the geometric theory has produced in the last thirty years.”
In late August, I visited Langlands in Montreal, in a small apartment that he owns with his wife, Charlotte Langlands, a sculptor. The day before, he had finished preparing for a lecture that he would be presenting in Turkey (in Turkish). Now that his Russian paper is published, Langlands says—though he has said this before—he is ready to stop working on mathematics and “go to other, more agreeable, things.”
The day we met, he had spent his first free morning in years reading a travelogue by the nineteenth-century German writer Theodor Fontane (in its original German). Having had time to rest and visit his family in British Columbia, Langlands was more energetic and conversational than he had been in Oslo a few months earlier—and a little more forgiving of his colleagues. He admitted he should probably have been more inviting of Frenkel’s questions.
In November, the Norwegian Academy of Science and Letters is co-hosting the Abel Conference in Minnesota, a three-day “mathematical celebration of Robert P. Langlands.” Langlands and Frenkel will both be in attendance. At the conference, a lecture will be presented explaining the basic premise of Langlands’s Russian paper and how it differs from the current geometric Langlands Program. Frenkel will be the one delivering it.