Any question I had, especially about the broader side of mathematics, he would answer clearly, often in a way that was more inspiring than anything I could have imagined.”ĭuring that time, Langlands laid the foundation for what eventually became his namesake program. “He was unlike any mathematician I had ever met. “I was essentially his first student, and I was very fortunate to have encountered him at that time,” Arthur says. (Langlands declined to be interviewed for this story.) Langlands was Arthur’s adviser at Yale University, where Arthur earned his Ph.D.

The Langlands program has always been a tantalizing dance with the unexpected, according to James Arthur, a mathematician at the University of Toronto. “I think that some of the most fascinating work that will come out of the next few decades is seeing consequences and manifestations of Langlands within parts of science where the interaction with this kind of pure mathematics may have been marginal up until now.” Overall Langlands remains mysterious, Rayan adds, and to know where it is headed, he wants to “see an understanding emerge of where these programs really come from.” A Puzzling Web “I think we’re only at the tip of the iceberg there,” he says. He has a hunch that, with time, links will be found between these programs and other areas as well. Researchers have already discovered strong links to physics, and Rayan and other scientists continue to explore new ones. Increasingly mathematicians are finding links between the original program-and its offshoot, geometric Langlands-and other fields of science. It has “many avatars, some of which are still open, some of which have been resolved in beautiful ways.”

Langlands is retired, but in recent years the project has sprouted into “almost its own mathematical field, with many disparate parts,” which are united by “a common wellspring of inspiration,” says Steven Rayan, a mathematician and mathematical physicist at the University of Saskatchewan.

Four years ago, he was awarded the Abel Prize, one of the most prestigious awards in mathematics, for his program, which was described as “visionary.” The program is named after Robert Langlands, a mathematician at the Institute for Advanced Study in Princeton, N.J. Edward Frenkel, a mathematician at the University of California, Berkeley, has even dubbed the Langlands program “a Grand Unified Theory of Mathematics.” It is such a force that some mathematicians say it-or some aspect of it-belongs in the esteemed ranks of the Millennium Prize Problems, a list of the top open questions in math. That program links seemingly disconnected subfields. Within mathematics, there is a vast and ever expanding web of conjectures, theorems and ideas called the Langlands program.

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