CHAPTER 278: CHAPTER 117: THE SILENT CONTRIBUTIONS OF THE MENTORS_2

Although the main lectures are still led by the primary leaders, Danis and Sam, the entire team is involved, including Professor Pan, with different emphasis on various aspects during the conferences.

Qiao Yu also astutely noticed that the man was the third most appeared in the scene...

Undoubtedly, for Qiao Yu, the discussion content of these conferences was very useful and indeed solved many of his doubts.

For instance, he gained a high-level understanding of the overall proof idea of the paper, so when reviewing the details again, many previously perplexing questions could be easily resolved.

Most importantly, it made him increasingly sure that his intuition might not be wrong.

When extending uniform equivalence results to the general case, it’s necessary to frequently use the local-global phenomena of algebraic geometry or category theory to locate and interact globally. Thus, if there’s a slight error in the local application, it would inevitably lead to a global error.

This might even indicate that the key theorem they proved—the applicability of the Ambidexterity theorem—might be limited in some specific cases.

We must know the last paper used this conclusion to generalize the conjecture to the general case. If the applicability of the most crucial Ambidexterity theorem is limited in the situation discussed in the paper, this proof of the Geometric Langlands Conjecture can only be declared a failure.

But proving this point is still not a simple matter for Qiao Yu.

Because the paper itself relies on specific axioms and settings, the results in high-order category theory are correct in a particular context, but if the axioms or category structures change, the theorem’s applicability might also be affected.

In fact, within the Geometric Langlands Program, some complex homological algebra problems handled using this theorem have already been successfully solved.

In layman’s terms, this paper reflects the results done within a constructed mathematical environment, relying on specific theoretical backgrounds and assumptions. To prove there’s an issue, Qiao Yu might need to find a way to prove that the entire constructed framework has logical flaws.

We must understand that in modern mathematics, the coherence of axiomatic systems and category theory frameworks is already highly rigorous. Any challenge or attempt to discover logical flaws must be based on more precise reasoning and innovative perspectives, making the task of challenging such constructions extremely difficult.

However, when trying to prove an error in mathematics, there is a most clever way, which is to construct a counterexample.

Counterexamples are extremely powerful tools in mathematics, which can directly show that a certain theorem or inference does not hold under specific conditions. Theoretically, as long as he can carefully design an algebraic geometric scenario within the logic framework built by the opposing side, where local objects cannot globally satisfy the requirements of the Ambidexterity theorem, he can achieve this aim.

If he could go further and discuss the reasons for axiom mismatches through this counterexample, such as tracing back the technical assumptions in the proofs to deduce the loopholes in the paper and provide a preliminary solution, he could probably once again become a star in the mathematical world...

Of course, this is still not a simple task.

In fact, it’s much more challenging than any problem Qiao Yu has encountered so far.

Anyway, a week after the training ended, it passed in plain dullness. He thought while reading books, reading papers, even while bathing, and sleeping, but still couldn’t construct a suitable counterexample.

However, while sitting on the high-speed train from Beijing back to Star City, Qiao Yu, as usual, sent his weekly work insights to Director Tian and the master across him.

"Respected Director Tian/Master: This week’s main work of mine is to delve into reading the proof of the Geometric Langlands Conjecture. The main gain of this week is that I have had some thoughts on one of the key conclusions, namely the Ambidexterity theorem, which I wish to report to you.

The Ambidexterity theorem played a very important role in this series of papers, especially when extending the conjecture from specific cases to a more general algebraic geometry context.

But as I further examined the structure of the theorem and its application in the proof of the Geometric Langlands Conjecture, I began to have some doubts about its applicability, especially when dealing with singularities or complex geometric situations.

According to my shallow understanding, the theorem relies on a certain equivalence between local and global objects, especially in frameworks of homological algebra and category theory, requiring the geometrically locally defined objects to maintain consistency globally.

This kind of local-global equivalence seems reasonable under a smooth geometric background. The paper also discussed some special cases, but I am considering if there are possible limitations in more complex situations, like the presence of extraordinary singularities on algebraic varieties?

Specifically, I suspect that the local structure near certain specific singularities may lead to some properties in homological algebra, such as local flatness or projectivity, being unable to be correctly globalized.

That is to say, if the Ambidexterity theorem must rely on such good behavior of local geometric structures, then on algebraic varieties with these specific singularities, might the applicability of the theorem be limited?

I have not yet found a specific counterexample, but in next week’s training activities, I intend to explore from the following two aspects: 1, whether there are extraordinary singularities that would affect the properties of local homological algebra, causing the theorem’s local-global equivalence to be disrupted.