Geometric Langlands Conjecture ... Proven?

[JMCx4]

From: Quanta Magazine

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.

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. ...

Read more at: Monumental Proof Settles Geometric Langlands Conjecture | Quanta Magazine

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[Hippasus]

This seems like a promising initiative, given how mathematics has drastically developed throughout history by bringing together different of its subareas. In this case, perhaps most prominently, geometry and number theory. It's like the connections are already there, or at least our minds are at built for looking for such connections. Edward Frenkel is a good proponent of the Langlands program. I saw him on one of Lex Fridman's podcasts. Admittedly, this stuff is way over my head at present, but here are a couple novice ideas that may or may not be relevant:

(1) Is not set theory (basically, logic with sets) already, amazingly, able to serve as a sort of foundation for all of mathematics since all of mathematics is expressible in its language, thus making Edward Frenkel's claim that the Langlands program is a "grand unified theory of mathematics" redundant? Perhaps it is a matter of 'which foundations' we seek to inquire about and develop. Could it be a sort of aesthetic choice? E.g. Joe prefers geometry, while Sue prefers algebra and analysis. I've heard of category theory (e.g. topoi) being considered as an alternative candidate for the foundations of mathematics, but I know next to nothing about this.

(2) I thought there was a claim from a famous mathematician in history (maybe Riemann, Cantor, or Fourier) to the effect that all mathematical expressions can be expressed by trigonometric functions. If true, this is mind-blowing to me.

Andre Weil is apparently in the historical background for the Langlands program. He was a constructive mathematician, from the early Twentieth Century, and thus in the minority from my contemporary standpoint.

 

[Fourier]

Robert Langlands is one of the best Canadian mathematicians ever. The program has been of incredible importance in mathematics for decades. His contributions were recognized in 2018 with the Abel Prize, one of the most prestigious prizes in mathematics.

 

[Fourier]

This seems like a promising initiative, given how mathematics has drastically developed throughout history by bringing together different of its subareas. In this case, perhaps most prominently, geometry and number theory. It's like the connections are already there, or at least our minds are at built for looking for such connections. Edward Frenkel is a good proponent of the Langlands program. I saw him on one of Lex Fridman's podcasts. Admittedly, this stuff is way over my head at present, but here are a couple novice ideas that may or may not be relevant:

(1) Is not set theory (basically, logic with sets) already, amazingly, able to serve as a sort of foundation for all of mathematics since all of mathematics is expressible in its language, thus making Edward Frenkel's claim that the Langlands program is a "grand unified theory of mathematics" redundant? Perhaps it is a matter of 'which foundations' we seek to inquire about and develop. Could it be a sort of aesthetic choice? E.g. Joe prefers geometry, while Sue prefers algebra and analysis. I've heard of category theory (e.g. topoi) being considered as an alternative candidate for the foundations of mathematics, but I know next to nothing about this.

(2) I thought there was a claim from a famous mathematician in history (maybe Riemann, Cantor, or Fourier) to the effect that all mathematical expressions can be expressed by trigonometric functions. If true, this is mind-blowing to me.

Andre Weil is apparently in the historical background for the Langlands program. He was a constructive mathematician, from the early Twentieth Century, and thus in the minority from my contemporary standpoint.

Just out of curiosity, where did you think you heard this. My sense is "mathematical expressions" might be functions, or more specifically "nice functions". In that case, it could well be something that either Fourier or possibly Riemann said. Fourier would of course be the natural one here, but Riemann did write his Habiltation on Fourier series.

Cantor of course is best known for foundations. But he did start out in harmonic analysis.

 

[Hippasus]

Just out of curiosity, where did you think you heard this. My sense is "mathematical expressions" might be functions, or more specifically "nice functions". In that case, it could well be something that either Fourier or possibly Riemann said. Fourier would of course be the natural one here, but Riemann did write his Habiltation on Fourier series.

Cantor of course is best known for foundations. But he did start out in harmonic analysis.

I think it might be Riemann, "On the representability of a function by means of a trigonometric series", SS3. "Not until January 1829, in Crelle's Journal, did Dirichlet's essay appear in which, for completely integrable functions without an infinite number of maxima and minima, he gave an entirely rigorous answer to the question of whether arbitrary functions could be represented by trigonometric series. . .

Aided by these. . .

. . . 3. where its value changes in sudden jumps, it adopts the average of the limit on either side.

A function that has the first two properties but not the third can obviously not be represented by a trigonometric series. For a trigonometric series that represented it apart from the discontinuities would diverge from it at the discontinuous points themselves. But whether and, if so, under what circumstances a function that does not fulfill the first two conditions can be represented by a trigonometric series, is left unanswered by this study.

Dirichlet's work provided a firm foundation for a whole host of important analytical investigations. Inasmuch he completely illumined (sp) the point on which Euler erred, he succeeded on settling a question that has occupied so many eminent mathematicians for more than seventy years (since 1753). In fact, it was completely settled for all cases in Nature with which we are concerned, for however great our ignorance of temporal-spatial changes in material forces and states in the infinitely small, even so we may safely assume that the function to which Dirichlet's investigations does not extend do not occur in Nature."

 

[Fourier]

I think it might be Riemann, "On the representability of a function by means of a trigonometric series", SS3. "Not until January 1829, in Crelle's Journal, did Dirichlet's essay appear in which, for completely integrable functions without an infinite number of maxima and minima, he gave an entirely rigorous answer to the question of whether arbitrary functions could be represented by trigonometric series. . .

Aided by these. . .

. . . 3. where its value changes in sudden jumps, it adopts the average of the limit on either side.

A function that has the first two properties but not the third can obviously not be represented by a trigonometric series. For a trigonometric series that represented it apart from the discontinuities would diverge from it at the discontinuous points themselves. But whether and, if so, under what circumstances a function that does not fulfill the first two conditions can be represented by a trigonometric series, is left unanswered by this study.

Dirichlet's work provided a firm foundation for a whole host of important analytical investigations. Inasmuch he completely illumined (sp) the point on which Euler erred, he succeeded on settling a question that has occupied so many eminent mathematicians for than seventy years (since 1753). In fact, it was completely settled for all cases in Nature with which we are concerned, for however great our ignorance of temporal-spatial changes in material forces and states in the infinitely small, even so we may safely assume that the function to which Dirichlet's investigations does not extend do not occur in Nature."

Thanks! As might have been obvious from my user name this is right up my ally. My research area is Abstract Harmonic Analysis. The "nice functions" I spoke of are essentially those being described by Riemann. He was correct that the case was settled for all functions in nature, or at least as nature was understood at the time since such functions were typically infinitely differentiable. But it turns out that this belief also led to some fundamental errors which eventually lead to the axiomatization of analysis and what we today see as the epsilon-delta approach.

 

[Hippasus]

Thank you so much. I am planning on taking real analysis in the fall.