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9 mathematicians, spanning more than 30 years, 5 papers totaling 800+ pages...

The geometric Langlands conjecture has finally been proved!

It is a geometrized version of the Langlands program.

Langlands ProgramIt is regarded as the largest single project in modern mathematics research and is called the "grand unified theory of mathematics". It proposes that the three independently developed branches of mathematics, number theory, algebraic geometry, and group representation theory, are actually closely related.

Fermat's Last Theorem was fully proved thanks to the application of the Langlands Program. Andrew Wiles's proof of the Langlands relation in number theory for a small number of functions solved a problem that had plagued the mathematical community for 300 years.

The geometric Langlands conjecture was proposed in the 1980s as a geometric version of the Langlands program. It provides a framework for applying number theory methods and concepts to geometric problems (and vice versa).

This conjecture can provide new ideas and tools for many unresolved problems in mathematics and physics, such as in the study of quantum field theory and string theory.

Therefore, when the geometric Langlands conjecture is proved, it will undoubtedly cause a sensation in the mathematical world.

Peter Scholze, a Fields Medal winner who primarily studies the Langlands Program, described the latest achievement as "the culmination of 30 years of hard work."

It’s really great to see it being resolved!

Alexander Beilinson, one of the founders of the Geometric Langlands Program, also said:

This proof is truly very beautiful and the best of its kind.

The study was led by Dennis Gaitsgory and Sam Raskin.

The 9-member team also includes Chinese scholarsChen Lin。

He is an assistant professor at the Qiu Chengtong Mathematical Sciences Center of Tsinghua University and won the IMO gold medal at the age of 15.

Geometry, the final link in the Langlands program

The Langlands Program was proposed in 1967.

30-year-old Princeton University professorRobert Langlands(Robert Langlands) sent a 17-page handwritten letter to André Weil, the creator of the "Rosetta Stone of mathematics", in which he outlined his vision.

(The "Rosetta Stone" here is a metaphor referring to an analogy between mathematical fields proposed by mathematician André Weil, which links three seemingly different mathematical fields: number theory, geometry, and function fields.)

Langlands writes that in “Rosetta Stone"In the field of number theory and functions, it is possible to create a generalization of Fourier analysis.

Fourier analysis, a framework for representing complex waveforms as smoothly oscillating trigonometric function waves, is a fundamental technique in modern telecommunications, signal processing, magnetic resonance imaging, and much of modern life.

Similar to the relationship between a function and its Fourier transform in Fourier analysis,The Langlands Program connects these three areas by establishing similar "correspondences" among them.

The Fourier transform converts back and forth between waves and spectra, and there are corresponding "waves" and "spectra" in the Langlands program.

The "wave" side is made up of some special functions, and the "spectrum" side is made up of some algebraic objects that mark the frequencies of the "wave":

• In number theory, functions are special functions defined on p-adic number fields or Adelian rings, and algebraic objects are representations of Galois groups or related groups;
• In geometry, a function is a characteristic layer (D-module) defined on a Riemann surface, and an algebraic object is a representation of the fundamental group of a Riemann surface on some algebraic group G;
• In the function field, functions are special functions defined on curves, and algebraic objects are representations of Galois groups or related groups.

Therefore, the Langlands Program provides a unified perspective that connects the three branches of mathematics: number theory, geometry, and function fields, and thus brings about a series of profound and extensive mathematical problems and conjectures.

Through the framework of the Langlands Program, many difficult problems in traditional number theory can be transformed into problems in representation theory or other fields, and thus solved with new perspectives and tools. The ideas and methods of the Langlands Program have been applied to many specific mathematical problems.

△Robert Langlands

For example, the proof of Fermat's Last Theorem borrowed ideas from the Langlands Program, linking elliptic curves and modular forms, and ultimately achieved success through these connections.

In addition to mathematics itself, the Langlands Program has also played an important role in other disciplines such as physics. For example, some ideas and methods of the Langlands Program have been applied in quantum field theory and string theory.

Among them, the geometric Langlands conjecture not only has wider applications and connections, but also provides powerful tools from a geometric perspective, so it is particularly important in the Langlands program.

However, the process of proving the geometric Langlands conjecture was also very difficult, spanning a total of 30 years, and the final proof work only began in 2013.

The core content of the proof is about the deep correspondence between self-similarity and symmetry on Riemann surfaces.

To explain it again using the model of Fourier analysis, mathematicians understood the "spectrum" side of the geometric Langlands conjecture very early on, but it took a long process to understand the "wave" side.

Even when Langlands first proposed this program, the geometric part was not included at all. It was not until the 1980s that mathematician Vladimir Drinfeld realized that it was possible to create a geometric version of the Langlands correspondence by replacing the eigenfunctions with eigenlayers.

The precise statement of the geometric Langlands conjecture only appeared in this century. In 2012, Dennis Gaitsgory and Dima Arinkin gave this statement in a paper of more than 150 pages.

Dennis and Alinkin pointed out that the core idea of ​​proving the geometric Langlands conjecture is to find an equivalence relation that connects the category of D-modules (solutions of differential equations on certain spaces) of G-bundles (fiber bundles on algebraic space G, whose fibers are copies of G) on algebraic curves X with the Ind-Coh category of the local system of the Langlands dual group ^ (including all Ind-cohomology objects), that is:

In 2013, Dennis wrote down a sketch of a proof of the geometric Langlands conjecture, but this sketch relied on many intermediate results that had not yet been proved. In the following years, Dennis and his collaborators worked to prove these results.

In 2020, Dennis began thinking about how to understand the contribution of each feature layer to the "white noise", an idea that later became a key part of the proof.

The "white noise" here refers to the Poincaré sheaf in conjunction with the Langlands conjecture, and the author's analogy is based on the sine wave in the Fourier transform.

In the spring of 2022, Sam Raskin and his student Joakim Færgeman proved that each feature layer contributes in some way to "white noise," a result that convinced Dennis that they would soon be able to complete the proof.

Starting from 2023, Dennis, Sam and seven other collaborators launched the final attack on the geometric Langlands conjecture. The final proof contained five papers with more than 800 pages and was published this year.

The first article is about the construction of functors. It requires constructing a geometric Langlands functor LG from automorphic to spectral direction in an environment with zero characteristics and proving its equivalence, that is, being able to establish a one-to-one correspondence between the two categories.

If this equivalence can be proved, it would show that the geometric Langlands conjecture holds.

The second paper studies the interaction between Kac-Moody localization and the global state, proving that the functor is indeed an equivalence functor under certain conditions, thus advancing the proof of the geometric Langlands conjecture.

The third article serves as a bridge, not only extending the known equivalence results to more general cases, but also providing key insights into the compatibility of geometric Langlands functors with constant-term functors through the Kac-Moody localization technique.

At the same time, by proving the compatibility of the geometric Langlands conjecture under reducible spectral parameters, this paper lays the foundation for further proving the geometric Langlands conjecture under irreducible spectral parameters.

In the fourth paper, the authors prove a key theorem, the Ambidexterity Theorem, which states that the left adjoint and right adjoint of LG-cusp (which can be viewed as the behavior of LG on a specific, smaller category) are isomorphic, which is an important step in proving that LG is an equivalence functor.

The final paper used this conclusion to generalize the conjecture to the general case, bringing an end to the protracted proof work.

Two generations of mathematicians work together to tackle the problem

The research team was led by Harvard University professor Dennis Gaitsgory and Yale University professor Sam Raskin.

Other authors, clockwise from left to right: Dario Beraldo, Lin Chen, Kevin Lin, Nick Rozenblyum, Joakim Færgeman, Justin Campbell, and Dima Arinkin.

△Image source: Quanta Magazine

It is worth noting that the research team includes Chinese scholars:Chen Lin。

Lin Chen is an assistant professor at the Qiu Chengtong Center for Mathematical Sciences at Tsinghua University. He received his bachelor's degree from Peking University in 2016 and his doctorate from Harvard University in 2021. He was awarded the Harvard 2020-2021 Excellence Scholarship.

He showed extraordinary mathematical talent when he was a teenager. At the age of 12, he entered the China Mathematical Olympiad (CMO) and won full marks. At the age of 15, he joined the national team and won the gold medal in the International Mathematical Olympiad (IMO).

Chen Lin has been studying the geometric Langlands program for a long time, and his connection with this direction came from Dennis Gatesgory.

Chen Lin revealed in a previous interview that he entered the field of geometric Langlands under the guidance of Dennis. Before his doctorate, he knew almost nothing about geometric representation theory, and learned a lot of basic knowledge under the guidance of Dennis.

After graduating with a Ph.D., Chen Lin has been involved in research projects with Dennis and other collaborators on the global categorized geometric Langlands conjecture.

After completing the proof of the conjecture and writing the paper, he will continue to think about the problem of local geometric Langlands.

In fact, the Langlands Program has attracted many Chinese mathematicians. Yun Zhiwei, Zhang Wei, Yuan Xinyi, and Zhu Xinwen, who are from the golden generation of Peking University, are also climbing this peak.

Reference Links:
[1]https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/
[2]http://www.mathchina.com/bbs/forum.php?mod=viewthread&tid=2060061