A Rosetta Stone for Mathematics

"In 1940, André Weil [Faculty (1958–76) and Emeritus (1976–98) in the School of Mathematics] wrote a letter to his sister, Simone, outlining his vision for translating between three distinct areas of mathematics. Eighty years later, it still animates many of the most exciting developments in the field."

In this letter, Weil "sketched his idea for a 'Rosetta stone' for mathematics. Following the example of the famous engraving by that same name—a trilingual text that made ancient Egyptian writing legible to Western readers through translation into Ancient Greek—Weil’s Rosetta stone linked three fields of mathematics: number theory, geometry, and, in the middle, the study of finite fields."

"Weil’s Rosetta stone has also guided progress in the Langlands program, a grand project to unify disparate fields of mathematics. The project began in 1967 when its founder, Robert Langlands [Professor Emeritus in the School of Mathematics], described his idea in a letter to Weil, expressing a desire to connect different branches of inquiry within number theory itself."

"Over the last few years, some of the most important advances in the Langlands program have involved translations between the original number-theory vision of Robert Langlands and the later geometric version. These translations follow the approaches set out in Weil’s Rosetta stone."

Such key contributions have been made by Laurent Fargues, von Neumann Fellow (2008) in the School of Mathematics, alongside his colleague Peter Scholze, who recently visited IAS to deliver the 2024 Emmy Noether Lectures. In 2021, Fargues and Scholze found a new way to connect number theory and geometry by defining the Fargues-Fontaine curve, a geometric object that can be used to prove a version of the Langlands correspondence.

Three other IAS Members, Edward Frenkel (1997, School of Mathematics), Pavel Etingof, (1996, School of Mathematics), and David Kazhdan (frequent Member and Visitor in the Schools of Mathematics and Natural Sciences), made another step forward by developing a new analytic framework for the Langlands correspondence that bridges the gap between the original intuitions and the geometric reformulation.

Read more at Quanta.