Selberg Trace Formula

Modern Mathematics and the Langlands Program

In his conjectures, now collectively known as the Langlands program, Robert Langlands drew on the work of Hermann Weyl (above), André Weil, and Harish-Chandra, among others with extensive ties to the Institute.

It has been said that the goals of modern mathematics are recon­struction and development.1 The unifying conjectures between number theory and representation theory that Robert Langlands, Professor Emeritus in the School of Mathematics, articulated in a letter to André Weil in 1967, continue a tradition at the Institute of advancing mathematical knowledge through the identification of problems central to the understanding of active areas or likely to become central in the future.

“Two striking qualities of mathematical concepts regarded as central are that they are simultaneously pregnant with possibilities for their own development and, so far as we can judge from a history of two and a half millennia, of permanent validity,” says Langlands. “In comparison with biology, above all with the theory of evolution, a fusion of biology and history, or with physics and its two enigmas, quantum theory and relativity theory, mathematics contributes only modestly to the intellectual architecture of mankind, but its central contributions have been lasting, one does not supersede another, it enlarges it.”2

In his conjectures, now collectively known as the Langlands program, Langlands drew on the work of Harish-Chandra, Atle Selberg, Goro Shimura, André Weil, and Hermann Weyl, among others with extensive ties to the Institute. 

Weyl, whose appointment to the Institute’s Faculty in 1933 followed those of Albert Einstein and Oswald Veblen, was a strong believer in the overall unity of mathematics, across disciplines and generations. Weyl had a major impact on the progress of the entire field of mathematics, as well as physics, where he was equally comfortable. His work spanned topology, differential geometry, Lie groups, representation theory, harmonic analysis, and analytic number theory, and extended into physics, including relativity, electromagnetism, and quantum mechanics. “For [Weyl] the best of the past was not forgotten,” notes Michael Atiyah, a former Institute Professor and Member, “but was subsumed and refined by the mathematics of the present.”3

The Fundamental Lemma: From Minor Irritant to Central Problem

The simplest case of the fundamental lemma counts points with alternating signs at various distances from the center of a certain tree-like structure. As depicted in the above image by former Member Bill Casselman, it counts 1, 1–3=–2, 1–3+6=4, 1–3+6–12=–8, etc. But this case is deceptively simple, and Bao Châu Ngô’s final proof required a huge range of sophisticated mathematical tools.

The fundamental lemma has been described as a gross understatement.1 “The curious thing is that it is called a lemma [a subsidiary proposition to be proved on the way to demonstrating a principal proposition]. It is a theorem,” says Andrew Wiles, a Visitor in the School of Mathematics and an Institute Trustee. “At first, it was thought to be a minor irritant, but it subsequently became clear that it was not a lemma but rather a central problem in the field.”

Robert Langlands, Professor Emeritus in the School of Mathematics, first introduced the fundamental lemma in 1979 in a lecture, “Les débuts d’une formule des traces stable,” at the École Normale Supérieure de Jeunes Filles and published in Publications Mathématiques de l’Université Paris VII.2 The goal of the lecture was the stabilization of the Selberg trace formula, but it also introduced the fundamental lemma, a technical device that links automorphic representations of different groups and the notion of closely related transfer factors that could transport automorphic forms. This led to the creation of a field of study that Diana Shelstad, a former Member in the School of Mathematics and Langlands’s student, eventually called “endoscopy.”