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https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics.

The proof represents the culmination of three decades of effort, said Peter Scholze, a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.”

The Langlands program, originated by Robert Langlands in the 1960s, is a vast generalization of Fourier analysis, a far-reaching framework in which complex waves are expressed in terms of smoothly oscillating sine waves. The Langlands program holds sway in three separate areas of mathematics: number theory, geometry and something called function fields. These three settings are connected by a web of analogies commonly called mathematics’ Rosetta stone.

The proof involves more than 800 pages spread over five papers. It was written by a team led by Dennis Gaitsgory (Scholze’s colleague at the Max Planck Institute) and Sam Raskin of Yale University.

https://people.mpim-bonn.mpg.de/gaitsgde/GLC/

https://en.wikipedia.org/wiki/Langlands_program

https://www.quantamagazine.org/what-is-the-langlands-program-20220601/

https://www.ias.edu/ideas/modern-mathematics-and-langlands-program

From the Quanta Magazine article

Now that the geometric Langlands researchers finally have their lengthy proof down on paper, Caraiani hopes they will have more time to talk to researchers on the number theory side. “It’s people who have very different ways of thinking about things, and there’s always a benefit if they manage to slow down and talk to each other and see the other’s perspective,” she said. It’s only a matter of time, she predicted, before the ideas from the new work permeate number theory.

As Ben-Zvi put it, “These results are so robust that once you get started, it’s hard to stop.”