CHAPTER 279: CHAPTER 117: THE SILENT CONTRIBUTIONS OF THE MENTORS_3

In the process of proving the Ambidexterity theorem, certain axiomatic structures in high-order category theory are involved. I want to further explore the performance of these category theory axioms in singular geometric situations, and whether there are some implicit assumptions that fail in more complex geometric contexts?

Although my idea may definitely seem naive to you, I think it has some exploratory value. The proof of the Geometric Langlands Conjecture is very complex, and the Ambidexterity theorem is a key conclusion within it. Any potential applicability issues could affect the validity of the proof.

Therefore, I hope to start with the local geometric structure at singular points, further verify the limitations of the theorem and potential problem areas. If you have a better idea, please let me know, your dearest student/grandson, for the first time this week, has felt the trouble of real hair loss."

This reflection was sent by Qiao Yu while sitting on a high-speed train to his mentor and grand-mentor; the person sitting next to him was Professor Zhou Liang, the leader of this IMO.

But actually, he had already edited this content last night and saved it on his phone. What he just did was copy and paste it, add the names, slightly modify the self-reference at the end, and then click the send button.

The main reason for doing this was to avoid being scolded by his mentor or grand-mentor again, saying he didn’t know the immensity of heaven and earth. After reading a paper for just a few days, he wants to find flaws in people’s work—this is quite possible.

The elderly are more receptive to him discovering loopholes in the course of studying, but his thoughts clearly hold the intention of nitpicking the paper.

But there is no way; honestly reporting step by step cannot reflect the seriousness of this problem. Right now, he really needs the help of these two big shots, hoping they can mobilize many brains to start from this direction and provide him with some constructive ideas.

It’s only natural to frankly express his ideas.

In short, he wants to fully utilize the resources around him and yet avoid bearing the responsibility that might arise from it.

Ultimately, he has been led astray by those two guys, Yu Yongjun and Gong Jiatao.

...

Yanbei University, Tian Yanzhen really didn’t expect Qiao Yu to suddenly send such a report today.

Due to participating in the training, Tian Yanzhen had actually considered that Qiao Yu could take a slight break this week. Who knew Qiao Yu not only didn’t rest but also showed him how frighteningly serious he could be!

Actually, the proof of the Geometric Langlands Conjecture hasn’t sparked much discussion outside the mathematics community.

Because the Langlands Program is too remote for ordinary people, it’s even less relatable than the Riemann Hypothesis, the N-S equation, and the like.

This is not to say the Langlands Program is necessarily more difficult than solving these world-class conjectures; the main point is that anything involving the unification of fundamental theories has an extremely high threshold.

For instance, the primary problem the Langlands Program needs to solve is to establish the bridge between Galois representation on algebraic number fields and autonomous forms. Just looking at the definition, you know that if you don’t spend years on algebraic geometry, number theory, and representation theory, you won’t even understand the question stem.

Really, if you don’t believe it, you can interview around the major schools of mathematics. Just an autonomous form alone can leave countless undergraduates and graduates learning until they’re exhausted, still half-understood, or for some, completely clueless.

If it’s the autonomous representation involved in the Langlands Program... that truly would be a "haha" moment. After all, autonomous forms are just an abstraction, while autonomous representations are a higher-level abstraction, describing how algebraic groups act on specific Hilbert or Banach spaces, where elements might be analytic functions or some special structures.

And the Geometric Langlands Program studies the correspondence between local systems on algebraic curves and the geometrization of autonomous forms. It merely replaces the number-theoretical objects in the classical Langlands Program with algebraic geometry objects.

The main research focus is geomorphizing abstract number theory problems so that they can be handled within the framework of algebraic geometry. This is not so much about solving specific problems, but rather providing mathematicians with valuable tools to solve more specific problems, with such a broad application potential.

This field mainly attracts mathematicians who hope to open new paths for number theory and algebraic geometry.

It can even be understood that the Geometric Langlands Program is similar to an important mathematical tool like calculus. Like the Langlands Program, calculus wasn’t invented to solve a particular problem directly but to construct a toolset for handling continuous change and limit problems.

So, the Geometric Langlands Program might completely change how future mathematicians handle certain problems, just as calculus changed how mathematicians of this era handle changes. Naturally, those interested in this field are mathematicians.

Because if someone indeed succeeds, it means that when researching number theory problems in the future, there will be a new framework that can be directly used, addressing classical number theory problems from a new dimension.

It might even, on some future day, become part of the essential curriculum of university mathematics, just like calculus.

Tian Yanzhen has also always been concerned about the development of the proof of the Geometric Langlands Conjecture. He is also optimistic about the results the research group achieved this time.

But who thought Qiao Yu would suddenly spring a big one on him.

His student suspects that the results someone has been researching for more than fifteen years and finally published have flaws?!