The first phase of IMO training was over. Although Qiao Yu planned to take a rest, he still honestly wrote a paper about his experience in reading the paper and sent it to Director Tian and Mr. Yuan from Huaqing. This time it was really not to let the two mentors know.

No matter how hard you work, you have a purpose.

Because he has been studying the geometric Langlands conjecture paper during the entire training period.

Director Tian didn't know if he would find someone to check his experience, but Qiao Yu believed that Mr. Yuan would casually forward it to Professor Pan. In this way, if he had any problems with his understanding, Professor Pan could point it out to him.

.

The main reason why he didn't send it directly to Professor Pan himself was because Qiao Yu felt embarrassed. Don't do to others what you don't want others to do to you.

If someone thinks of a question every day, just send it to him directly... And he can summarize several questions every day, and this continues for nearly half a month. Qiao Yu feels that he has blocked this person directly.

Although Professor Pan did not block him, he had clearly felt that Professor Huaqing's messages were getting slower and slower.

Anyway, in the first two days when I joined him, he was the quickest in replying to messages. Almost every time I had a question, I would reply immediately. After that, the reply to messages became slower and slower. During the training period, there were even questions he sent the day before. After a full time

It took Professor Pan a day and a half to reply.

Qiao Yu didn't consider whether the questions he asked were getting more and more tricky, and Professor Pan needed to discuss it with someone before he dared to reply. Moreover, Professor Pan didn't tell him that he was not in Huaqing at all, but was going abroad for a meeting.

Went.

Qiao Yu just sensitively felt that Professor Pan had probably changed and was beginning to annoy him. It was difficult to ask this question directly, so he had to make a roundabout way. There was nothing to say in the next week, so he continued to read books and papers.

read.

It was just that Qiao Yu was supposed to read the paper on prime numbers before, but instead read the more than 800-page paper on the geometric Langlands conjecture. Reading and learning at the same time would undoubtedly slow down the speed of reading the paper.

In the eyes of normal professors, the books I read this week are beginning to look unsystematic. But this one is what Qiao Yu is best at, finding books to read as needed.

There are quite a lot of books to read about the Langlands Program of Geometry. Qiao Yu felt tired before because he had no direction, but now that he has a goal to strive for, this kind of task does not feel so tiring anymore.

Moreover, the training time schedule is not too tight. The collective life also ensures that Qiao Yu can go to bed on time at 11 o'clock every day and get up on time at 6:30 in the morning. The sleep he lost some time ago was fully made up during the training camp.

By the way, there is one more thing worth mentioning.

Yu Yongjun also brought Gong Jiatao into his friend group.

There is no way, the two of them have been getting along best these days. However, as the group leader, when Qiao Yu saw these two people in the same group, he decided to talk less in the group in the future. After all, it is easy to hide from the obvious.

, it’s hard to guard against darkness.

Yu Yongjun is a standard bright and coquettish character, with extremely strong brain jumping ability, but he only talks but does not do all kinds of shameless coquettish operations.

Gong Jiatao is naturally a secret coward. He looks like a dog to ordinary people, but he is very liberal in playing various tricks. We will wait for him to do it first. As for Yu Wei...

Originally, Qiao Yu thought that he was the only kind and good person in the group, but he was a bit withdrawn. But now it seems that this guy can lie without blushing and his heart does not beat, and he looks very sinister.

At this time, Qiao Yu suspected that the reason why these people wanted to participate in the competition instead of taking the college entrance examination normally was probably because if they took the college entrance examination, everyone would find out that based on their ideological and political level, they should be arrested and shot immediately.

Of course, the feeling is mutual.

What Qiao Yu didn't know was that when Yu Yongjun and Gong Jiatao often chatted in private, they actually had similar views... For example, non-human beings like Qiao Yu should not participate in math competitions with them, and their destination should be sent to a certain place. A high-end biological laboratory conducts direct biopsy research.

The friendship formed during the competition is so pure and beautiful. Of course, this does not mean that the two sides will really be on the same page.

In fact, everyone has been learning from each other. It's like Qiao Yu thinks that he should learn from Yu Yongjun and Gong Jiatao's shamelessness, and Gong Jiatao and Yu Yongjun have agreed that after returning to school this time, they must find some classmates and play with them. A game where you can solve problems while playing.

They even tried it secretly in private and found that if they replaced the Olympiad questions with ordinary high school practice questions, they could still handle it. Of course, the rules of the competition would have to be slightly modified, and they would not be able to play against those who were very good at playing games.

But these don't seem to be problems. For ordinary students in the high school they live in, just being able to keep up with most of their classmates requires them to do their best, and there isn't much left to devote to games.

After all, those who are really qualified to waste some time on games during their studies these days have to be primary school students...

But their reasons are also very good. Only by regaining the self-esteem that has been stepped on by others can they regain their confidence and achieve good results in international competitions. From this point of view, among the four friends, they can write in one stroke Yu Wei is probably the only one who can be said to be kind.

But Qiao Yu didn't care.

The only thing that made him feel sad was that recently, even Director Tian and Grandpa Shi across from him were getting slower and slower in replying to his messages.

Well, at the beginning of the semester, the bosses are all very busy, Qiao Yu comforted himself like this.

But there are still gains. For example, Professor Pan, who was getting slower and slower in replying to messages, suddenly one day sent an address directly to Qiao Yu and gave him an account password. After logging in, Qiao Yu discovered the latest few messages in it. These videos are all videos of various recent seminars on the geometric Langlands conjecture.

Although the lectures were still given by the two main leaders, Dennis and Sam, they also included everyone from the entire team, including Professor Pan, and the focus of the demonstrations in the several lectures was also different.

Qiao Yu also noticed keenly that the man appeared in the third most scenes...

There is no doubt that for Qiao Yu, the discussions in these seminars were very useful and indeed answered many of his questions.

For example, if you have a high-level understanding of the overall proof idea of ​​the paper, then when you read through the details of the paper, you will be able to solve many problems that you did not understand before.

Most importantly, it made him more certain that his intuition might be right.

When extending consistent equivalence results to general situations, it is necessary to frequently use local-global phenomena in algebraic geometry or category theory to locate and interact with the global. Then if there are some minor errors in local calls, it will inevitably

Will cause a global error.

This may even lead to the key theorem proved by the authors - that the applicability of Ambidexterity may be limited in certain situations.

You must know that the last paper uses this conclusion to extend the conjecture to general situations. If the applicability of the most critical Ambidexterity theorem is limited in the situation discussed in the paper, then this time the proof of the geometric Langlands conjecture is only

Can declare failure.

But Qiao Yu wants to prove this is still not an easy thing.

Because the paper itself relies on specific axioms and assumptions, the results in higher-order category theory are correct in a specific context, but if the axioms or category structure change, the applicability of the theorems may also be affected.

Even in the geometric Langlands program, some complex homological algebraic problems treated by this theorem have been successfully solved.

In words that ordinary people can understand, this paper is the result of an environment constructed by mathematicians themselves, relying on specific theoretical background and assumptions. To prove that there is a problem, Qiao Yu may need to find a way to prove the construction.

The entire framework presented has logical holes.

We must know that in modern mathematics, the autonomy of axiomatic systems and category theory frameworks is inherently rigorous. Any work that questions or attempts to find logical loopholes must be based on more rigorous reasoning and innovative perspectives, which makes questioning this structure

The task is extremely difficult.

But when you want to prove an error in mathematics, there is a trickiest way, which is to construct a counterexample.

Counterexamples are a very powerful tool in mathematics, which can directly show that a certain theorem or inference is not valid under certain conditions. In theory, as long as he can carefully design an algebraic geometry situation within the logical framework established by the other party, and let

This goal can be achieved if local objects cannot globally satisfy the requirements of the Ambidexterity theorem.

If he can go one step further and use this counterexample to discuss the reasons for the axiom mismatch, for example, through the technical assumptions in the backtracking proof, to deduce the loopholes in this paper, and give a preliminary solution, then he can probably become a

A star in mathematics...

Of course, this is still not a simple matter.

In fact, it is much more difficult than any problem Qiao Yu has encountered so far.

Anyway, the week after the training camp passed so uneventfully. He thought about it while reading books and papers, and even thought about it while taking a shower and sleeping, but he still couldn't construct a suitable counterexample.

However, while sitting on the high-speed train from the capital back to Star City, Qiao Yu still shared his work experience this week with Director Tian and Grandpa Shi across from him as usual.

"Dear Master Tian/Grandpa Shi: My main job this week is still to read in depth about the proof of the geometric Langlands conjecture. The main gain this week is that I have some thoughts on one of the key conclusions, namely the Ambidexterity theorem. Special report to you.

The Ambidexterity Theorem plays a very important role in this series of papers, especially in generalizing the conjecture from specific situations to a more general algebraic geometry background.

But as I further looked at 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 situations involving singular points or complex geometry.

According to my superficial understanding, this theorem relies on a certain equivalence of local and global objects, especially in the framework of homologous algebra and category theory, which requires locally defined geometric objects to be globally consistent.

This kind of local-global equivalence seems reasonable in the context of smooth geometry. The paper also discusses some special cases, but when I am thinking about some more complex cases, such as the case where an algebraic variety contains extraordinary singular points ,Are there possible limitations?

Specifically, I suspect that local structure near some specific singular points may cause certain properties in homology algebra, such as local flatness or projectiveness, to not globalize correctly.

That is, if the Ambidexterity theorem must rely on this good behavior of local geometric structures, is there a limit to the applicability of the theorem to algebraic varieties where such specific singular points exist?

I haven't found a specific counterexample yet, but in next week's training event I plan to think deeply from the following two aspects: 1. Whether there are extraordinary singular points that will affect the properties of local homology algebra and trigger local problems of the theorem. —Global equivalence is broken.

2. The proof of the Ambidexterity theorem involves certain axiomatic structures in higher-order category theory. I would like to further explore the performance of these category theory axioms in singular geometric situations, and whether there are some implicit assumptions that cannot be used in more complex situations. Established in geometric background?

Although my ideas may certainly seem naive to you, I think they have some exploratory value. The proof of the geometric Langlands conjecture is very complicated, and the Ambidexterity theorem is a key conclusion in it, and any potential applicability issues may Have an impact on the validity of the proof.

So I hope to start with the local geometric structure at the singular point to further verify the limitations and potential problems of the theorem. If you have a better idea, please tell me quickly, your dearest student/grandson, this week For the first time, I felt the pain of losing hair."

This experience was sent by Qiao Yu to his mentor and grandpa Shi while he was sitting on the high-speed train. The person sitting next to him was Professor Zhou Liang, the leader of this IMO team.

But in fact, he had already edited this content last night and saved it on his phone. What he just did was copy, paste and add the names of the people, then slightly change the self-identification at the end, and then click the send button.

The main reason for doing this is to avoid being asked by the tutor or the master to give him a lecture again, saying that he does not know how high the sky is. It has only been a few days since I read the paper, and I want to find loopholes in others - this is very possible.

The old man is more accepting of the fact that he found loopholes in the process of learning, and his thinking was obviously based on the idea of ​​​​causing faults in other people's papers.

But there is no way. An honest and step-by-step report cannot reflect the seriousness of this problem. He is in great need of help from the two big guys now. It is best to mobilize many brains to start from this direction and give him some construction. Inspired by sexual thoughts.

Naturally, he must tell his thoughts truthfully.

To put it bluntly, I want to make full use of the resources around me, but I don’t want to bear the responsibilities caused by it. After all, I was led astray by two guys, Yu Yongjun and Gong Jiatao.

..

At Yanbei University, Tian Yan really didn't expect that Qiao Yu would suddenly send such a report today.

Because he had to participate in the training camp, Tian Yanzhen had actually agreed that Qiao Yu could take a break this week. Who would have thought that not only did Qiao Yu not take a break, but he also showed him how scary it is to be serious!

In fact, the proof of the geometric Langlands conjecture has not triggered much discussion outside the mathematical community.

Because the Langlands Program is too far away for ordinary people, it is not even as friendly as the Riemann Hypothesis, N-S equations and other things.

This is not to say that the Langlands Program is necessarily more difficult than solving these world-class conjectures. The main thing is that the threshold for anything involving the unity of basic theory is extremely high.
To be continued...