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Selected from quantamagazine

By Erica Klarreich

Compiled by Machine Heart

Editor: Panda

After three decades of hard work, mathematicians have successfully demonstrated the main parts of a grand mathematical vision called the Langlands program.

A team of nine mathematicians has successfully proved the Geometric Langlands Conjecture, one of the most widespread paradigms in modern mathematics.

The proof is the culmination of three decades of hard work, said Peter Scholze, a renowned mathematician at the Max Planck Institute for Mathematics who was not involved in the work. “It’s great to see it solved.”

The Langlands Program was proposed by Robert Langlands in the 1960s. It is a broad generalization of Fourier analysis, a far-reaching framework for representing complex waves as multiple smoothly oscillating sinusoids. The Langlands Program has important roots in three different areas of mathematics: number theory, geometry, and the so-called function field. These three areas are connected by a network of analogies that has been called the "Rosetta Stone" of mathematics.

Now, a series of papers prove the Langlands conjecture for the geometrical columns of this Rosetta Stone: https://people.mpim-bonn.mpg.de/gaitsgde/GLC/

“No other field has ever had this kind of comprehensive and robust evidence,” says David Ben-Zvi of the University of Texas at Austin.

“This is beautiful mathematics, the most beautiful kind,” said Alexander Beilinson, one of the main pioneers of the geometric version of the Langlands program.

The proof consists of five papers totaling more than 800 pages. They come from a team led by Dennis Gaitsgory (Max Planck Institute) and Sam Raskin (Yale University).

Gaitsgory has spent the past 30 years working on proving the geometric Langlands conjecture. The proof builds on decades of research by him and his collaborators. Vincent Lafforgue of Grenoble Alpes University likens these advances to a "rising sea"; he says it's like the work of Alexander Grothendieck, a great 20th-century mathematician, who solved difficult problems by creating a rising sea of ​​ideas.

Dennis Gaitsgory (left) and Sam Raskin (right), who led a nine-member team that proved the geometric Langlands conjecture.

It will take some time to verify their new proof, but many mathematicians say they believe the core idea is correct. "The theory has such good internal consistency that it's hard to believe it's wrong," Lafforgue said.

In the years leading up to the proof, the team created more than one path to the heart of the problem. “The understanding they arrived at was so rich and so broad that they surrounded the problem from all directions,” he says. “There was no escape.”

Grand Unified Theory

In 1967, Robert Langlands, then a 30-year-old Princeton professor, laid out his vision in a 17-page handwritten letter to André Weil, the creator of the Rosetta Stone. Langlands wrote that it would be possible to create a generalized version of Fourier analysis in the number theory and function field fields of the Rosetta Stone, which would have amazing scope and power.

In classic Fourier analysis, a correspondence is created between two different ways of thinking about a wave pattern (like a sound wave) using a process called the Fourier transform. On one side of this correspondence are the waves themselves. (We call this the wave side.) This includes simple sine waves (pure tones in acoustics) as well as complex waves made up of multiple sine waves. On the other side of this correspondence is the spectrum of cosine waves - the pitches in acoustics. (Mathematicians call this the spectral side.)

The Fourier transform goes back and forth between these two sides. In one direction, it breaks a wave down into its set of frequencies; in the other direction, it reconstructs the wave from its component frequencies. This ability to transform in both directions enables countless applications - without it, we wouldn't have modern telecommunications, signal processing, magnetic resonance imaging, or many other necessities of modern life.

Langlands proposed that similar transformations occur in the number theory and function field fields of the Rosetta Stone, but the waves and frequencies here are more complicated.

In the video below, Rutgers University mathematician Alex Kontorovich takes us through this mathematical continent and into the amazing symmetries at the heart of the Langlands Program.

Video source: https://www.youtube.com/watch?v=_bJeKUosqoY

In each of these fields, there is a waveform consisting of a set of special functions that resemble repeating waves. The purest of these special functions is called an eigenfunction, and it acts like a sine wave. Each eigenfunction has a characteristic frequency. However, while the frequency of a sine wave is a single number, the frequency of an eigenfunction is an infinite list of numbers.

Then there’s the spectral side. This consists of objects from number theory; Langlands thought that these objects mark the frequency spectrum of the characteristic function. He proposed that there’s a Fourier transform-like process that connects this wave side to the spectral side. “It’s a bit of a magic trick,” Ben-Zvi said. “It’s not something we can predict in advance without any reason.”

Waves and their frequency labels come from very different data domains, so proving a correspondence between them is bound to be very rewarding. For example, in the 1990s, a proof of a number-theoretic Langlands correspondence for a relatively small set of functions allowed Andrew Wiles and Richard Taylor to prove Fermat’s Last Theorem — one of the most famous unproven problems in mathematics, and one that has been grappling with for three centuries.

The Langlands program is seen as a "grand unified theory of mathematics," says Edward Frenkel of the University of California, Berkeley. Yet even as mathematicians have worked to prove larger and larger parts of Langlands' vision, they are well aware that it is incomplete. The relationship between waves and frequency labels doesn't seem to fit in the geometric fields of this Rosetta Stone.

Grain of sand

It was from Langlands’ work that mathematicians had an idea of ​​what the spectral side of the geometric Langlands correspondence looked like. The third column of Weil’s Rosetta Stone (geometry) involved compact Riemann surfaces, including spheres, donuts, and porous donuts. For a given Riemann surface there is a corresponding object, called a fundamental group, which keeps track of the different forms of loops that can wrap around the surface.

Mathematicians conjecture that the spectral side corresponding to geometric Langlands should be composed of specific distilled forms of the fundamental group, which are also called representations of the fundamental group.

If the Langlands correspondence is to be reflected in the geometry field of the Rosetta Stone, then each representation of the fundamental group of a Riemann surface should be a frequency label - but a frequency label of what?

Mathematicians couldn’t find any set of eigenfunctions whose frequencies seemed to mark the fundamental group representation. Then in the 1980s, Vladimir Drinfeld, now at the University of Chicago, realized that it might be possible to create a geometric Langlands correspondence by replacing the eigenfunctions with more complex objects called eigensheafs — but at the time, he knew only a few ways to build eigensheafs.

Sheafs are so much deeper than functions that number theorists didn’t know what to make of Langlands’s geometric cousin. But the geometric Langlands program, for all its esotericity, has one big advantage over its number-theoretic counterpart. In geometric Langlands, the frequencies of the eigensheets are controlled by points on a Riemann surface, a sphere or a doughnut that looks very similar up close. But in number-theoretic Langlands, the frequencies are controlled by prime numbers, and each prime has its own unique properties. Mathematicians didn’t know “how to get from one prime to another in a nice way,” says Ana Caraiani, a number theorist at Imperial College London.

Riemann surfaces play an important role in physics, especially in conformal field theory, which governs the behavior of subatomic particles in certain force fields. In the early 1990s, Beilinson and Drinfeld showed how you could use conformal field theory to construct some particularly nice character layers.

This connection to conformal field theory got Beilinson and Drinfeld thinking about how to construct a Fourier analysis for the sheaf. “It’s like a grain of sand that triggers crystallization,” Ben-Zvi said.

Beilinson and Drinfeld offer a rich vision of how the geometric Langlands correspondence should work. It’s not just that each representation of the fundamental group should label the frequencies of a feature layer. The correspondence should also respect important relations on both sides, they argue, a prospect that Beilinson and Drinfeld call “the best hope.”

In the mid-1990s, Beilinson presented this evolving research landscape in a series of lectures at Tel Aviv University. Gaitsgory, then a graduate student there, absorbed every word. “I was like a newly hatched duckling, acquiring an imprinting behavior,” he recalls.

In the 30 years since, the geometric Langlands conjecture has been the main driving force of Gaitsgory's mathematical career. "It's been a constant work for many years, getting closer and closer to the goal, developing different tools," he said.

Rising Sea

Beilinson and Drinfeld stated their conjecture only loosely, and it turned out that they had oversimplified the way the relationship in their “best hope” was supposed to work. In 2012, Gaitsgory and Dima Arinkin of the University of Wisconsin-Madison figured out how to turn that “best hope” into a precise conjecture.

The following year, Gaitsgory wrote an outline of a possible way to prove the geometric Langlands conjecture. The outline relied on a number of intermediate statements, many of which had not yet been proved. Gaitsgory and his collaborators set out to prove them.

Over the next few years, Gaitsgory and Nick Rozenblyum of the University of Toronto wrote two books on layers that together totaled nearly 1,000 pages. In the two volumes, the geometric Langlands program was mentioned only once. “But the purpose was to lay the foundations, and we used those foundations extensively later,” Gaitsgory said.

In 2020, Gaitsgory suddenly found himself without a schedule. “I spent three months lying in bed, just thinking,” he said. Those thoughts eventually led to a paper (with six authors). Although the paper focused on the function field column of the Langlands program, it also contained “a seed” that later became a key component of proving the geometric Langlands conjecture: a method for understanding how feature layers promote so-called “white noise.”

Photo of the other seven researchers. Clockwise from left: Dario Beraldo, Lin Chen, Kevin Lin, Nick Rozenblyum, Joakim Færgeman, Justin Campbell and Dima Arinkin.

In classical signal processing, sound waves can be constructed from sine waves, whose frequencies correspond to the pitches in the sound. It's not enough to know which pitches a sound contains - you also need to know how loud each pitch is. This information allows you to write the sound as a combination of sine waves: just start with a sine wave with amplitude 1, then multiply that sine wave by the appropriate loudness factor, and add those sine waves together. The sum of all the different sine waves with amplitude 1 is what we often call "white noise."

In the world of the geometric Langlands program, feature sheaves act like sine waves. Gaitsgory and his collaborators identified something called a Poincaré sheaf that seems to act like white noise. But it wasn’t clear to the researchers whether every feature sheaf could be represented in a Poincaré sheaf, let alone whether they all had the same amplitude.

In the spring of 2022, Raskin and his graduate student Joakim Færgeman showed how to use the ideas from the six-author paper to prove that every characteristic layer is indeed representable in the Poincare layer. “After Sam’s and Joakim’s papers, I was pretty sure we could do it in a short time,” Gaitsgory said of the proof of the geometric Langlands conjecture.

The researchers needed to prove that all feature layers contribute equally to the Poincare layer and that the fundamental group representation labels the frequencies of these feature layers. They realized that the hardest part was dealing with this fundamental group representation: an irreducible representation.

These solutions to irreducible representations came at a time when Raskin’s personal life was in turmoil. A few weeks after he and Færgeman posted their paper online, Raskin had to rush his pregnant wife to the hospital before returning home to take his son to kindergarten for the first time. Raskin’s wife stayed in the hospital for six weeks until their second child was born. During this time, Raskin’s life was in a state of turmoil—an endless scramble between home, his son’s school, and the hospital to keep his son on track. “My whole life was about cars and taking care of people,” he says.

He called Gaitsgory to discuss math during his drive. Towards the end of the first of those weeks, Raskin realized he could reduce the irreducible representation problem to proving three facts that were then within reach. “That was a magical time for me,” he said, adding that his personal life “was filled with anxiety and fear about the future. Math was something that needed grounding and meditation for me to get out of that anxiety.”

By early 2023, Gaitsgory and Raskin, along with Arinkin, Rozenblyum, Færgeman and four other researchers, had completed a proof of Beilinson and Drinfeld’s “best hope,” revised by Gaitsgory and Arinkin. (The others are Dario Beraldo of University College London, Lin Chen of Tsinghua University, and Justin Campbell and Kevin Lin of the University of Chicago.) The team spent another year writing up the proof. They posted it online in February. Although the papers follow the outline Gaitsgory laid out in 2013, they simplify Gaitsgory’s approach and improve on it in many ways. “A lot of smart people contributed a lot of new ideas to this incredible achievement,” Lafforgue says.

“They didn’t just prove it,” Ben-Zvi said. “They built a whole world around it.”

Further coast

For Gaitsgory, the realization of this decades-long dream is far from the end of the story. There are many further puzzles for mathematicians to solve—to explore its connection with quantum physics more deeply, to extend the result to Riemann surfaces with holes, to figure out its impact on other columns of the Rosetta Stone. “It feels (at least to me) more like we have chiseled off a big rock, but we are still far from the core,” Gaitsgory wrote in an email.

Researchers working on the other two columns are now eager to translate the proof. “The fact that one of the major fragments has been solved should have a major impact on the overall study of the Langlands counterpart,” Ben-Zvi said.

But not everything carries over — for example, there is no equivalent in number theory and function field settings to the ideas of conformal field theory, which allow researchers to build special layers of features in geometric settings. Many of the ideas in the proof will require some laborious adjustments before they can be used in other fields. It’s not clear whether we can “transfer these ideas to a different context where they were not originally thought to be used,” said Tony Feng of Berkeley.

But many researchers are optimistic that this rising sea of ​​ideas will eventually spill over into other fields. “It will penetrate all the barriers between disciplines,” Ben-Zvi says.

Over the past decade, researchers have begun to discover connections between the geometric field and the other two fields. “If (the geometric Langlands conjecture) had been successfully proved 10 years ago, the results would have been very different,” Feng said. “People would not have realized that its impact could extend beyond the (geometric Langlands) community.”

Gaitsgory, Raskin, and their collaborators have made some progress in translating the geometric Langlands proof into the field of function fields. (Raskin hints that some of the discoveries Gaitsgory and Raskin made during the latter’s long drive “are still to be revealed.”) If successful, the result could be a far more precise version of function-field Langlands than mathematicians have known or even guessed before.

Most translations from the geometric field to the number-theoretic field go through the function field. But in 2021, Laurent Fargues and Scholze of the Jussieu Institute of Mathematics in Paris designed a so-called wormhole that brings ideas from the geometric field directly to a part of the Langlands program in number theory.

“I’m definitely someone who wants to translate these geometric Langlands proofs,” Scholze says. Considering that the rising sea contains thousands of pages of text, that’s no mean feat. “I’m a couple of papers behind at the moment,” Scholze says, “and I’m trying to read up on what they did in 2010 or so.”

Now that the geometric Langlands researchers have finally published their lengthy argument, Caraiani hopes they’ll have more time to discuss it with number theorists. “People think in very different ways,” she says. “It’s always good if they can slow down and talk to each other and understand each other’s perspectives.” She predicts that it’s only a matter of time before the ideas from this new result spread to number theory.

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

Original link: https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/