School of Mathematics

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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

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Of Historical Note

By John Wheeler 

The paper by Niels Bohr and John Wheeler on the mechanism of nuclear fission appeared in the Physical Review of September 1, 1939, the same day the war began.

The following excerpt is from remarks given by John Archibald Wheeler on March 27, 2000, in connection with the play Copenhagen by Michael Frayn. Wheeler was a Professor of Physics at Princeton University from 1938 until his retirement in 1976 and a Member of the Institute’s School of Mathematics (prior to the founding of the School of Natural Sciences) in the spring of 1937, when it was still temporarily housed in Fine Hall (now Jones Hall) at Princeton University. Niels Bohr, who had a twenty-year association with the Institute, first visited in the academic year 1938–39, when the Institute completed Fuld Hall. For more about Bohr and his relationship with Albert Einstein, one of the Institute’s first Professors, see the Spring 2009 Institute Letter.

If two such great thinkers as Bohr and Einstein, who had such a high regard for each other, could be brought together for a prolonged period, would not something emerge of great value to all of us? This thought and this hope animated the guiding spirits of the Princeton Institute for Advanced Study to invite Niels Bohr to come as a guest of the Institute for the entire spring semester of 1939. However, four days before Bohr boarded his America-bound ship, he learned from Otto Robert Frisch that Frisch and his aunt Lisa Meitner had solid evidence that a neutron splits the nucleus of uranium. As he crossed the Atlantic, Bohr’s vision turned more and more from the problem of quantum mechanics to the problems of nuclear physics. So January and February, March and April of 1939 saw him working, discussing, calculating, and writing, day after day, not with Einstein on quantum physics as intended, but with me on the nuclear physics of fission. Yes, of course, there were meetings Bohr had with Einstein but they were occasional and did not lead to the big push it takes to formulate a solid well-argued position. No. Fission, and what it meant and how it differed from one nucleus to another, and what those differences offered in the way of using the nucleus for a chain reaction stood at the center of our attention. . . .

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