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Robert Langlands: Far-Reaching Mathematics

By Kelly Devine Thomas Published 2007

Cliff Moore

Robert Langlands’s visionary program has had a deep influence in mathematics and parts of theoretical physics.

Born on October 6, 1936, in British Columbia, Robert Langlands grew up in a small town where his father owned a building supply store. “When I was a child I liked to add and subtract,” says Langlands. “In our store, my mother worked. And I remember competing with her. We would tally lumber; she would do it on the adding machine and I would do it in my head.”

He enrolled at the University of British Columbia at age 16, returning home during the summers to work with his father, hauling lumber and cement. At the university, he was drawn to mathematics over physics, particularly algebraic computations. “In the course of my life I have spent much time with algebraic computations. I never minded,” says Langlands. “They’re often irritating things, because it is clear what the answer must be, but the answer turns out otherwise. It may take days to find the small error.”

Langlands received his Ph.D. from Yale University and taught at Princeton and Yale Universities before joining the Faculty of the Institute in 1972. At Princeton University, he was encouraged to teach a course in class field theory by his colleague Salomon Bochner, who drew him to the attention of the Institute’s late Atle Selberg. “I had an hour’s conversation with Selberg and he explained the details of some of his ideas,” says Langlands. “That was my first professional contact with a really first-class mathematician. It was an experience I will never forget.”

But it was the Institute’s Harish-Chandra with whom Langlands felt a particular affinity. “His papers were among the first—Selberg’s papers and then Harish-Chandra’s—that I studied very carefully, that I actually worked with, that I actually used in what I was doing,” says Langlands. “Harish-Chandra was the one who proposed me for the Institute. The first time I didn’t meet with his colleagues’ approval, and the second time I did.”

Of his first meeting in the early 1960s with the Institute’s André Weil, to whom he would address a letter five years later suggesting conjectures that would become collectively known as the Langlands program, Langlands’s recalls: “He came into my office in the old Fine Hall, threw his foot up over the arm of the chair, and he started to talk to me. Somehow he had heard of me as a promising young person and he’d come to me, which is striking. I was never someone who ever learned to be silent, and I remember telling him my ideas, and they were foolish ideas, I must say. I was already inclined to formulate conjectures and some of them weren’t bad, but some of them were foolishly wrong, a result of inexperience.”

In 1967, Langlands ran into Weil at a lecture given by Shiing-Chen Chern. Langlands had been making calculations when he came across some L-functions that he liked. He told Weil what he was thinking and Weil suggested he write it down for him. Langlands went home and wrote a seventeen-page handwritten letter, showing it to Harish-Chandra before sending it on to Weil. A typed copy of the letter, made at Weil’s request for easier reading, circulated widely among mathematicians in the late 1960s and 70s, and for more than three decades now, mathematicians have been working on its conjectures.

In his letter, Langlands proposed a grand unifying theory that relates seemingly unrelated concepts in number theory, algebraic geometry, and the theory of automorphic forms. “There were some fine points that were right that rather surprise me to this day,” says Langlands. “There was evidence that these L-functions were good but that they would have these consequences for algebraic number theory was by no means certain.” As William Casselman observed, Langlands’s astounding insight “has provided a whole generation of mathematicians working in automorphic forms and representation theory with a seemingly unlimited expanse of deep, interesting, and above all approachable problems to
work away on.”

Some aspects of the Langlands program have been proven, such as Laurent Lafforgue’s proof of the “Langlands conjecture for function fields,” which Lafforgue presented in a series of lectures at the Institute in 1999 and for which he won a Fields Medal. Other aspects have led to proofs of seemingly unrelated theorems, such as Andrew Wiles’s 1994 proof of Fermat’s last theorem. “Nobody understood really before Wiles the implications for Fermat’s theorem. That was something that will have an enormous influence,” says Langlands. “Proposals of interest only to mathematicians, and then only to a small circle of mathematicians, now had potential appeal to anyone with even a slight interest in scientific ideas.”

The geometric Langlands program, created by Vladimir Drinfeld and collaborators, is particularly rich for implications in theoretical physics, especially string theory. Last year, Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, who says his understanding of the Langlands program is limited (see below) wrote a 225-page paper on the relation of part of the geometric Langlands program to physics. Asked if he understands Witten’s work in relation to his program, Langlands replies, “I would say at the moment no. I think I would like to. At first, I wanted to understand it out of curiosity but now I think there are other reasons for wanting to understand it.”

Despite tremendous progress, the core problem (functoriality) of the Langlands program remains unsolved, according to Langlands. Moreover, Langlands observes, “[Alexander] Grothendieck left behind some unrealized ideas. There are connections between these ideas and my proposals.What I like to think, although I may be fooling myself, is that the two will someday merge.”

Courtesy of the Institute for Advanced Study Archives

Langlands conversing with former Member Philip Kutzko

In recent years, Langlands has spent some of his time studying a field unrelated to the Langlands program—lattice models of statistical physics and the attendant conformal invariance. His interest in languages continues unabated, particularly his study of Russian, which he laments, remains “one of the unrequited intellectual loves of my life.” Now and again, he likes to write something about the history of mathematics. “I am rather fascinated by mathematics as a chapter in the history of ideas,” says Langlands.

In the 1999–2000 academic year, Langlands presented a series of Institute talks, “The Practice of Mathematics,” aimed at explaining several central mathematical problems to a general audience. The series, for which Langlands drew from the title of a poem by William Butler Yeats, “Beautiful, Lofty Things,” for inspiration, grew from eight to sixteen lectures over the course of the year. “It grew partially because it was fun to do,” says Langlands, who notes that the lectures were inspired in part by the late Clifford Geertz, Professor in the School of Social Science. “He would occasionally ask me about mathematics,” explains Langlands. “When I started, I had him implicitly in mind as a member of the audience, as the audience.”

Over the years Langlands has been a strong supporter, financially and otherwise, of the intellectual and cultural life of the Institute, engaging in discourse with his colleagues in the other Schools and attending its lectures and concerts with his wife Charlotte, to whom he has been married for 51 years. They have four grown children and several grandchildren. “At the Institute, there is the possibility for a richer intellectual life than is often possible at a university,” says Langlands.

Arriving at the Institute at age thirty-six and being granted the freedom to pursue his curiosity and unfettered time to think, Langlands says, has been a true blessing. “I wish often that I’d done better, but nonetheless, it’s a real reason to be grateful.”

Edward Witten on Geometric Langlands

The Langlands program is incredibly vast and far-reaching. The deepest aspect of it, as far as we know, involves the number theory setting where Langlands started close to forty years ago. However, the Langlands program has all kinds of manifestations. The part that I have tried to understand personally is the “geometric’’ form of the Langlands program, where some of the ideas are converted from number theory into statements in geometry. For a long time, mathematicians working on the geometric Langlands program have made great use of ideas from mathematical physics—notably an area called conformal field theory that is important both in condensed matter physics and in string theory. But the physics ideas were always rearranged in ways that—to a physicist—looked strange. This bothered me a lot for years, really for decades. I felt that if physics-based ideas are relevant to the geometric Langlands program, then it should be possible to reformulate the geometric Langlands program in terms that would be more recognizable to a physicist. Eventually, after a lot of false starts, I did have some success with this. Despite all the hard work, I personally only understand a tiny bit of the Langlands program. I think, however, that this probably puts me in the majority among researchers who work on it. It is such a vast subject that few can really have an overview. And where it will ultimately lead, it is way too soon to say.––Edward Witten, Charles Simonyi Professor in the School of Natural Sciences

After thirty-five years at the Institute for Advanced Study, Robert Langlands, whose visionary work, known as the Langlands program, has had a deep influence across a broad sweep of mathematics and parts of theoretical physics, retired in July as Hermann Weyl Professor in the School of Mathematics, becoming Professor Emeritus.

Langlands’s profound insights in number theory and representation theory include the formulation of general principles relating automorphic forms and algebraic number theory; the introduction of a general class of L-functions; the construction of a general theory of Eisenstein series; the introduction of techniques for dealing with particular cases of the Artin conjecture (that proved to be of use in the proof of Fermat’s theorem); the introduction of endoscopy; and the development of techniques for relating the zeta functions of Shimura varieties to automorphic L-functions.

Awarded the 2007 Shaw Prize in Mathematical Sciences, Langlands’s other honors include the Frederic Esser Nemmers Prize in Mathematics (2006); the Grande Médaille d’Or (Gold Medal) of the French Academy of Sciences (2000); the Wolf Prize in Mathematics (jointly with Andrew Wiles, 1996); the inaugural National Academy of Sciences Award in Mathematics (1988); the Common Wealth Award (1984); and the American Mathematical Society’s Cole Prize (1982).

Kelly Devine Thomas is Editorial Director of the Institute for Advanced Study.

Published in The Institute Letter Fall 2007