I Monumental Proof Settles Geometric Langlands Conjecture

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A group of nine mathematicians has successfully proved the geometric Langlands conjecture, a significant milestone in modern mathematics that connects number theory, geometry, and function fields. This proof, spanning over 800 pages across five papers, culminates three decades of research and was led by Dennis Gaitsgory and Sam Raskin. The Langlands program, initiated by Robert Langlands in the 1960s, serves as a foundational framework for understanding complex mathematical relationships. Experts anticipate that the new findings will foster collaboration between researchers in geometry and number theory, enhancing the exchange of ideas. The achievement is viewed as a monumental step forward in the field of mathematics.
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group 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.
https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719

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.”
 
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Wow, this is pretty monumental. It's like trying to span the Grand Canyon: You have half of the bridge across it built and now need to build the other connecting half.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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