Monumental Proof Settles Geometric Langlands Conjecture | Quanta Magazine
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Monumental Proof Settles Geometric Langlands Conjecture
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Langlands program
Monumental Proof Settles Geometric Langlands Conjecture
By
Erica Klarreich
July 19, 2024
In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.
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Nan Cao for Quanta Magazine
Introduction
By Erica Klarreich
Contributing Correspondent
July 19, 2024
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geometry
group theory
harmonic analysis
Langlands program
mathematics
number theory
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A 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.
Now, a new set of papers has settled the Langlands conjecture in the geometric column of the Rosetta stone. “In none of the [other] settings has a result as comprehensive and as powerful been proved,” said David Ben-Zvi of the University of Texas, Austin.
“It is beautiful mathematics, the best of its kind,” said Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program.
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.
Gaitsgory has dedicated the past 30 years to proving the geometric Langlands conjecture. Over the decades, he and his collaborators have developed a massive body of work on which the new proof rests. Vincent Lafforgue, of Grenoble Alps University, likened these advances to a “rising sea,” in the spirit of the preeminent 20th-century mathematician Alexander Grothendieck, who spoke of tackling hard problems by creating a gradually rising sea of ideas around them.
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Dennis Gaitsgory (left) and Sam Raskin led the nine-person team that proved the geometric Langlands conjecture.
From left: Natasha Bershadsky; Charlotte Krontiris
It will take mathematicians a while to digest the new work, but many have expressed confidence that the core ideas are correct. “The theory has a lot of internal consistencies, so it’s difficult to believe there could be a mistake,” Lafforgue said.
In the years leading up to the proof, the research team created not one but many routes into the heart of the problem, Ben-Zvi said. “The understanding that they’ve developed is so rich and so broad, they’ve encircled the problem from every direction,” he said. “It had no way to escape.”
A Grand Unified Theory
In 1967, Robert Langlands, then a 30-year-old professor at Princeton University, laid out his vision in a handwritten 17-page letter to André Weil, the originator of the Rosetta stone. Langlands wrote that in the number theory and function field columns of the Rosetta stone, it might be possible to create a generalization of Fourier analysis with startling scope and power.
A Rosetta Stone for Mathematics
number theory
A Rosetta Stone for Mathematics
May 6, 2024
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In classical Fourier analysis, a procedure called the Fourier transform creates a correspondence between two different ways of thinking about the graph of a wave (such as a sound wave). On one side of the correspondence are the waves themselves. (We’ll call this the wave side.) These include both simple sine waves (which in acoustics are pure tones) and more complicated waves that are combinations of sine waves. On the other side of the correspondence is the spectrum of frequencies of the sine...