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In classical signal processing, sound waves get built up out of sine waves whose frequencies correspond to the pitches contained in the sound. It’s not enough to know which pitches the sound contains — you must also know how loud each pitch is. That information allows you to write your sound as a combination of sine waves: just start with the sine waves of amplitude 1, then multiply each sine wave by an appropriate loudness factor before adding the sine waves together. The sum of all the different amplitude-1 sine waves is what we commonly refer to as white noise.
In the world of the geometric Langlands program, eigensheaves are supposed to play the role of sine waves. Gaitsgory and his collaborators had identified something called the Poincaré sheaf that seemed to be serving the role of white noise. But the researchers didn’t know whether each eigensheaf is even represented in the Poincaré sheaf, let alone whether they all have the same amplitude.
In the spring of 2022, Raskin, together with his graduate student Joakim Færgeman, showed how to use the ideas in the six-author paper to prove that each eigensheaf does contribute to the Poincaré sheaf. “After Sam’s and Joakim’s paper, I was certain we’ll do it within a short period of time,” Gaitsgory said of proving the geometric Langlands conjecture.
The researchers needed to show that all the eigensheaves make equal contributions to the Poincaré sheaf, and that the fundamental-group representations label the frequencies of these eigensheaves. The trickiest part, they came to realize, was handling representations of the fundamental group called irreducible representations.
The solution for these irreducible representations came to Raskin at a moment when his personal life was filled with chaos. A few weeks after he and Færgeman posted their paper online, Raskin had to rush his pregnant wife to the hospital, then return home to take his son to his first day of kindergarten. Raskin’s wife remained in the hospital until the birth of their second child six weeks later, and during this time Raskin’s life revolved around keeping life normal for his son and driving in endless loops between home, his son’s school and the hospital. “My whole life was the car and taking care of people,” he said.
He took to calling Gaitsgory on his drives to talk math. By the end of the first of those weeks, Raskin had realized that he could reduce the problem of irreducible representations to proving three facts that were all within reach. “For me it was this amazing period,” he said. His personal life was “filled with anxiety and dread about the future. For me, math is always this very grounding and meditative thing that takes me out of that kind of anxiety.”
By early 2023, Gaitsgory and Raskin, together with Arinkin, Rozenblyum, Færgeman and four other researchers, had a complete proof of Beilinson and Drinfeld’s “best hope,” as modified by Gaitsgory and Arinkin. (The other researchers are Dario Beraldo of University College London, Lin Chen of Tsinghua University in Beijing, and Justin Campbell and Kevin Lin of the University of Chicago.) It would take the team another year to write up the proof, which they posted online in February. While the papers follow aspects of the outline Gaitsgory developed back in 2013, they both simplify his approach and go beyond it in many ways. “Very bright people contributed a lot of new ideas to this crowning achievement,” Lafforgue said.
“It wasn’t just that they went and proved it,” Ben-Zvi said. “They developed whole worlds around it.”
Further Shores
For Gaitsgory, the fulfillment of his decades-long dream is far from the end of the story. A host of further challenges await mathematicians — exploring the connection to quantum physics more deeply, extending the result to Riemann surfaces with punctures, and figuring out the implications for the other columns of the Rosetta stone. “It feels (at least to me) more like that one piece of a big rock has been chipped off, but we are still far from the core,” Gaitsgory wrote in an email.
Researchers working in the other two columns are now eager to translate what they can. “The fact that one of the major pieces has fallen should have major repercussions throughout the Langlands correspondence,” Ben-Zvi said.
Not everything can carry over — for instance, in the number theory and function field settings, there is no counterpart to the conformal field theory ideas that enabled researchers to construct special eigensheaves in the geometric setting. Much of the proof will need serious adjustment before it can be made to work in the other two columns, warned Tony Feng of Berkeley. It remains to be seen, he said, whether “we can even transport the ideas to a different context where it was not designed to work.”
But many researchers are optimistic that the rising sea of ideas will eventually reach these other domains. “It’s going to seep through all the barriers between subjects,” Ben-Zvi said.
In the past decade, researchers have started turning up unexpected connections between the geometric column and the other two. “If [the geometric Langlands conjecture] had been proved 10 years ago, then the results would be very different,” Feng said. “It wouldn’t have been appreciated that it could potentially have ramifications outside [the geometric Langlands] community.”
Gaitsgory, Raskin and their collaborators have already made progress on translating their geometric Langlands proof to the function field column. (Some of the discoveries Gaitsgory and Raskin made on the latter’s long car drives are “still to come,” Raskin hinted.) If successful, this translation will prove a much more precise version of function field Langlands than mathematicians knew or even conjectured before now.
Most translations from the geometry column to the number theory column pass through function fields along the way. But in 2021, Laurent Fargues, of the Mathematics Institute of Jussieu in Paris, and Scholze devised what Scholze called a wormhole that carries ideas from the geometric column directly over to a part of the number theory Langlands program.
“I’m definitely one of the people who are now trying to translate all this geometric Langlands stuff,” Scholze said. With the rising sea having spilled over into thousands of pages of text, that is no easy matter. “I’m currently a few papers behind,” Scholze said, “trying to read what they did in around 2010.”
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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