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Chapter 121 Congratulations, you have seen some beautiful scenery

"...we know that in the study of algebraic curves, the choice of ridges and the construction of automorphic forms are key...the key to the biggest problem is that the Galois representation may only be valid in a specific p-adic environment. When we establish a p-adic number field..."

Professor Everton's report was still drilling into Qiao Yu's ears, but Qiao Yu's mind was already completely filled with all kinds of lines.

Just a little bit, just a little bit...

It's not these things...

Because it has been demonstrated in the paper...

The author has used many restrictions very carefully to ensure the correctness of the results and to find a new approach.

and many more……

When finding faults, one should not be bound by existing ideas. As long as it is within the framework set up by the other party and does not exceed the framework, he can also define it freely...

An unusual singularity, but not an ordinary singular structure...

"Got it!" Qiao Yu suddenly shouted excitedly.

The conference room suddenly became quiet, and countless eyes were focused on the front row.

The professors in the back row could only see the backs of heads, but the bigwigs sitting in the first row and Professor Everton on the stage stared with confusion at the young man who suddenly shouted out with an excited look on his face.

After all, this is really rude, even if Qiao Yu is just a minor.

Especially Pan Jingyuan, who sat closest to Qiao Yu, was frightened by the sound just now and shivered. He was totally confused...

Qiao Yu also reacted quickly: Fuck, I’m in trouble!

So the boy jumped up from his chair as if by reflex, and immediately apologized: "I'm very sorry, Professor Everton and all the teachers, it's really that what Professor Everton just said was too wonderful!

It gave me a lot of inspiration, and suddenly made me think of a very special counterexample, which seems to be able to prove that there are local-global errors in the proof of the Ambidexterity Theorem in the series of papers recently published by Professor Dennis and Professor Sam on ArXiv.

I was so excited about this idea, and I was so absorbed in thinking that I forgot I was attending the lecture. I'm sorry, I'm so sorry!"

After saying that, Qiao Yu bowed slightly to Professor Everton on the podium.

After Qiao Yu finished speaking, the meeting room became even quieter.

Many relevant professors were invited to participate in this series of seminars, but whether it was Yuan Zhengxin or Tian Yanzhen, they naturally could not tell the truth when they invited these professors that it was because their own students wanted to verify the papers recently published by Dennis and Sam.

After all, that would be too offensive.

We just invited people here in the name of discussing the geometric Langlands conjecture.

Professors from abroad, China, Huaqing, Yanbei and Shuangdan University gathered together to discuss the topic.

Although everyone knew that after several exchanges, the five papers that have been in the spotlight recently would definitely be involved in the end, but probably no one expected that in the first lecture, the youngest guy in the conference room would jump up and intend to destroy the other side's place.

Yuan Zhengxin didn't expect this either, so his eyes were also fixed on Qiao Yu.

It is a good thing to have inspiration, and it is understandable. It is even normal to be excited, but it is questionable whether the counterexamples in your mind are valuable!

Well, Professor Everton on the podium naturally didn't expect this. He subconsciously glanced at Pan Jingyuan next to Qiao Yu.

Professor Dennis had introduced this student to him before, so of course he knew him.

Last night he also invited Pan Jingyuan to discuss some ideas about the proof process between him and his supervisor.

He couldn't quite understand why the young boy next to Pan Jingyuan suddenly stood up and said such words.

Professor Pan's expression at this moment was indeed very strange...

But he didn't look at him, but stared blankly at Qiao Yu who was standing next to him.

Professor Everton looked at Qiao Yu, thought for a moment, and said, "I can accept your apology, but you have to convince me with what you just thought of. If you can make me think that this idea is really great, I am not only willing to forgive you, but also to thank you for making this lecture have some different value."

Qiao Yu subconsciously turned his head to look at his master who was sitting in the first row.

Mr. Yuan glared at him first, then nodded imperceptibly. Qiao Yu was not polite and immediately said, "Okay, Professor Everton, this is what I think. First, let's assume..."

Everton waved at Qiao Yu and said, "Kid, it's not polite to stand down there and talk about your ideas. Come up and speak on the stage. I think your brain definitely hasn't prepared a PPT for you, so..."

As he said this, Everton looked back and smiled: "There happens to be a blackboard here."

Since the big boss said so and the master also agreed, Qiao Yu, who has always been courageous, immediately left his seat and walked to the podium.

It just so happens that my mind is a little confused, and I can use the explanation to clear up my thoughts.

"Teachers, here's what I think. First, let's assume a type of singular point of an algebraic variety. This type of singular point is different from the known singular structures, such as cusps, nodes, or cone-shaped singular points. It has a complex ridge-like extension globally.

The point is that its local geometry is conjugated with other points at the far end of the algebraic variety, even non-adjacent singular points. So, first we need to define its local behavior. Assuming that it is in A3 or a higher-dimensional geometric space, its characteristic equation should be..."

As he spoke, Qiao Yu picked up the chalk and wrote a line of equation on the blackboard: "f(x, y, z)=z^2x^3y^2+sin(xyz)".

After writing it, Qiao Yu took a step back, calculated silently in his mind for a moment, and then continued: "I believe everyone has already seen that this equation has a local ridge-like limit structure somewhere near the point (0, 0, 0)."

"Well, its conjugation relationship is manifested in the fact that when the singular points on the generation number cluster are P1 and P2, respectively, they have local ridge-like singular point structures, and their local geometric properties influence each other through a nonlinear homology mapping.

Obviously, this means that the local mode structure of the singular point P1 will depend on the local properties of another remote singular point P2. Note that this conjugation relationship cannot be inferred by simple local geometric observations..."

At this point, Qiao Yu's voice suddenly stopped...

The audience was equally silent, but the reactions were varied.

Some people have frowned, picked up paper and pen, and started calculating on the manuscript paper they carried with them; some are still listening attentively; and some are still in shock, watching the development of the situation.

However, Everton on the stage was staring at the equation written by Qiao Yu, reading it with great interest.

As for Pan Jingyuan in the audience, he was definitely the one with the deepest frown on his face. As the person who was most familiar with the series of papers at the scene, he had vaguely guessed Qiao Yu's general idea, but he had not yet figured out what method Qiao Yu would use to break the deadlock.

But he reacted quickly, glanced at Yuan Zhengxin who was also looking at Qiao Yu seriously, and then took out his phone...

The old man might not necessarily take out the live video, so he might as well record one himself first.

……

At this time, Qiao Yu's brain was also thinking quickly.

Although he found the key point, he still had to design an algebraic cluster background based on the framework constructed in the five-part series of papers.

It happened suddenly, and he just had a direction. He had to design this background in a hurry, which tested his ability to perform on the spot. Fortunately, it would take some time to calculate the equation he threw out.

Qiao Yu didn't bother to care what others thought of him now. Since no one was urging him, he just thought silently.

After thinking for five minutes, Qiao Yu, who was standing in front of the blackboard, suddenly picked up the chalk again. This time he did not speak, but started writing directly on the blackboard.

“Consider a high-dimensional algebraic variety X, defined as an algebraic variety of the form: X = {(x, y, z, w) ∈ A4 | z^2x^3y^2+sin(xyz)+w^5=0}.”

After writing, Qiao Yu took a step back again and began to calculate quickly in his mind. After another minute, he spoke: "According to the explanation just now, you should be able to find that the algebraic cluster X has such a singular point near (0, 0, 0, 0), and it has a conjugate relationship with the far end of the algebraic cluster through the variable w.

More specifically, P1/P2 are two singular points with the same structure. Yes, that's right! Then we need to consider the spine structure under the p-adic framework mentioned by Professor Everton.

We can see that near the singular points P1 and P2 of X, the local homology algebraic structure manifests itself in two ways. First, near P1, the flatness and projectivity of the local p-adic module M1 are extended to the far end P2 through the ridge network, so that the module that appears flat near point P1 no longer maintains projectivity.

Second, because of their conjugate structure, the singular points P1 and P2 influence each other through nonlinear homological algebraic relations, which leads to abnormal behavior of the Ext group near the ridge network. That is, the local module structure near P1 cannot be correctly globalized, which undoubtedly destroys the local-global equivalence.

That’s all I can think of for now. Of course, this is just the first step. We will need some time to analyze the local modular structure. Then we will introduce higher-order category theory and derive the failure of functors.

If you have read the first paper by Dennis and Professor Sam on the proof of the geometric Langlands conjecture, you should know that if you can prove that the local to global derived functor cannot satisfy the consistency in the homology category, that is: RHomC(M1, M2)LHomC(M1, M2)
It is sufficient to show that the local-to-global equivalence of the Ambidexterity Theorem fails in this context. The local homology algebraic flatness condition assumed by the theorem is likely to no longer hold in the presence of this type of structure. "

After saying that, Qiao Yu put down the chalk, clapped his hands, and carefully looked at the two formulas he left on the blackboard. Then he turned around and faced all the professors in the audience for the first time, as well as Professor Everton standing next to him.

Then Qiao Yu realized belatedly that everyone at the scene, yes, everyone, every one of them, was in an extremely strange state.

Among them was his great-grandmaster, Mr. Yuan, and Professor Everton on the stage.

You can't say they were all dumbfounded, because everyone had different expressions, but they all seemed to be frozen.

This reminded him of an old Japanese TV show, in which the protagonist had the ability to stop time, and then only he could move...

Of course, what Qiao Yu thought of was Dinosaur Express Ks-X, but we can't say that people who think in the wrong way have unhealthy thoughts. After all, there are too many movies like this...

Anyway, the reactions of the crowd made Qiao Yu a little uneasy, and he couldn't help but cough twice: "Ahem, um... Professor Everton, I'll go down first?"

"Oh... um... okay... you can go down first, kid. By the way, if I'm not mistaken, you are Professor Yuan's grandson. Your name is Qiao Yu, right?" Professor Everton asked after coming back to his senses.

"Yes, Professor Everton." Qiao Yu, who had just walked down from the podium, turned around and answered.

"Sixteen years old?" Everton asked again.

Qiao Yu nodded.

Professor Everton pursed his lips, glanced in the direction of Yuan Zhengxin, then gave a thumbs up and said, "You are great!"

Then he watched Qiao Yu sit back in his seat, glanced at his PPT, and continued, "Then let's continue... Forget it, let's first roughly calculate the counterexample that Professor Yuan's grandson Qiao Yu just gave us..."

Having said that, Everton simply turned around, picked up the chalk, and began to do some simple deductions on the blackboard.

After a moment, Everton suddenly turned around and asked, "Qiao Yu, this is the first type of singularity you defined. Maybe you should give it a name, otherwise I don't know how to represent it."

"Uh..." Qiao Yu was stunned, obviously he had never thought about this question.

"Well, it seems you haven't thought about it yet, so let's call it JOE point." Everton shrugged, then turned his head and continued to deduce on the blackboard.

Qiao Yu took a deep breath and didn't watch the old professor's deduction process. He had already calculated it in his mind and thought it should be fine. So he just secretly turned his head to glance at Professor Pan Jingyuan beside him.

To be honest, the person he felt most sorry for doing this was Professor Pan.

People have patiently guided him for so long, and he felt a little embarrassed when he turned around and proved that the paper that people had participated in researching for many years had flaws.

Who knew that while he was secretly looking at someone else, Professor Pan turned his head and looked at him openly. Their eyes met, and Qiao Yu felt even more embarrassed.

He smiled awkwardly and said in a low voice: "Well, Professor Pan, please don't be depressed. My imagination may not be correct."

"Hmm..." Pan Jingyuan responded in a low voice. Qiao Yu was confused and didn't know what it meant, but his sitting posture and expression became obedient at the same time, and there was no sign of arrogance on his face.

Again, no matter how much armor the instructor and his master gave him, it would only increase his magic defense, but not his physical attacks. Brother Pan was right next to him, so he had to be more honest in this situation.

It is a well-known fact that when honest people get angry, they can be very scary.

……

Professor Everton stood on the podium and calculated silently. The time had exceeded the originally scheduled time. Gradually, someone in the audience might have thought of something. They stood up and walked to the podium to discuss with Professor Everton. Then, more content written in different handwritings appeared on the blackboard.

With the first, there is the second...

Soon, the podium was surrounded by people, and even Pan Jingyuan, who was next to him, walked up to the podium.

Qiao Yu was stunned...

What the hell, is this lecture over yet? Can he leave now?
So Qiao Yu looked at Mr. Yuan on the other side.

The old man was watching the discussion on the stage with a smile, then he seemed to realize something and turned his head to look at Qiao Yu who was sitting restlessly. He probably guessed what Qiao Yu was thinking, so he waved at Qiao Yu slightly.

It doesn’t really matter whether Qiao Yu’s sudden idea is right or wrong.

The fact that the bigwigs in these research fields have taken this seriously and started discussing it says a lot.

Especially the JOE spot.

When Professor Everton asked that question in public, Qiao Yu had already achieved a huge success today.

The old man was very happy at this time, even happier than when he had achieved his goals.

Qiao Yu also quickly came over.

"Did you really just think of it?" the old man asked.

Qiao Yu nodded repeatedly, with an innocent expression like a child.

"Really! Grand Master, I just suddenly thought of it."

Mr. Yuan smiled and said, "Okay, go back and think about the complete proof process first, and you'd better sort it out quickly. Don't let your inspiration go to waste. We will choose a day for you to give a lecture next time. We will set up the stage for you, and you can go up and speak boldly. Don't be afraid."

Qiao Yu nodded immediately.

He had nothing to be afraid of in the first place, and now with Mr. Yuan's words, he felt even more relaxed.

"Also, no matter whether your idea is right or wrong, please set a time to invite Professor Pan to dinner. I'll make the appointment for you. Thank Professor Pan."

"No problem, Master, I'll go back first." Qiao Yu nodded repeatedly, then stood up and sneaked away.

The old man was right, the rest of the proof would have to be done by him personally.

Now that he was called up by the professors, Qiao Yu had nothing more to say. As he had just explained, he had only constructed a counterexample. He still had to think carefully about the specific proof process. As he said, mathematical proofs must not have logical loopholes.

……

Qiao Yu left, but the discussion and even debate on the podium continued. There were also professors who did not go to the podium to participate in the discussion. They watched Qiao Yu sneak up to Mr. Yuan. After the two of them whispered a few words, the child crouched and sneaked away, probably with envy in his heart.

Other professors were using their mobile phones and various chat tools to discuss the counterexample that Qiao Yu had just designed in another way.

In short, the conference room was in a mess at this time.

Especially on the podium, the voices were getting louder and louder.

In addition to the two equations left by Qiao Yu on the blackboard, the derivation process next to them had been erased and written again, and had been changed several times.

Under the stage, Mr. Yuan just watched quietly with a smile on his face. No one knew what he was thinking about. It seemed as if he was reminiscing about the good old days.

Pan Jingyuan also stood on the podium, watching silently. He went up to the podium because there were too many people, covering the blackboard. Occasionally, someone would chat with him in a low voice, but he just shook his head slightly and didn't say anything.

In fact, to be honest, he was probably the one who could best follow Qiao Yu's thoughts at the scene. After all, Qiao Yu had been discussing a series of issues with him, but he really didn't want to talk at all at this moment.

It's not only bitter that six years of hard work was discovered by a sixteen-year-old boy, but the key is that this boy only took about a month to do it!
This suddenly made him feel that all the work he had done before had lost its sense of accomplishment.

Oh, I’m so tired!
……

After walking out of the conference room, Qiao Yu did not think about the mood of the professors who were still there. His mind was not idle either. On the one hand, he had to think about how to make a complete proof, and on the other hand, he had to think about what methods could solve the problem he raised.

Qiao Yu believes that if he finds a counterexample to this type of problem, he can find many counterexamples to invalidate the theorem. For example, after opening up his mind, the structure of the local coordinate ring of a similar singular point can produce infinite levels of high-order perturbations in the conventional decomposition.

Then the p-adic modules or ridge structures show fractal characteristics at these singular points, and their complexity cannot be described by finitely generated local modules. Therefore, the derived culprits cannot satisfy the consistency in the homology category in any way.

If there is no equivalence, the theorem cannot hold.

Of course, we can continue to restrict the conditions and throw out such counterexamples. But once we do this, the influence of the so-called proof of the geometric Langlands conjecture will be greatly reduced. After all, the theorem cannot be fully established in the context of smooth geometry, so its application is naturally greatly limited.

However, patching this vulnerability is much more difficult than finding it.

This is actually the same as in reality, destruction is always easier than construction.

Qiao Yu didn’t know what to do for the time being, but he could sort out some ideas.

If you can find a way, you can send it out at the right time.

In short, asking yourself questions and then giving answers is a way of learning.

Of course, Qiao Yu was still very excited at this time.

He has already won half of this peak showdown game. The next question is thrown to the opposing team to prove the geometric Langlands conjecture. After he publishes the complete proof process, it will depend on whether the other team can solve the problem within the time limit they expect.

It's a pity that in reality this kind of game can only be turn-based, and is not as exciting as games like FPS or MOBA.

But it doesn’t really matter.

Excitedly returning to his room, Qiao Yu stretched and was about to turn on the computer when there was a knock on the door.

Qiao Yu stood up again and opened the door. Senior Brother Chen was standing outside.

That would have been fine, but Qiao Yu could see that his senior brother's look at him was also very strange, and he didn't say anything, just stared at him blankly, which was even weirder.

Qiao Yu subconsciously touched his face, which was smooth and tender. Although he did use facial cleanser today, he shouldn't be so handsome that even his senior brothers would be surprised, right?
"Um, Senior Brother Chen, there's nothing on my face, right?" Qiao Yu asked.

Chen Zhuoyang shook his head and said, "No, Junior Brother, you are in the limelight again today!"

That tone was one of admiration that could not be concealed at all.

Qiao Yu immediately retorted, "No, Brother. Although we have a good relationship, if you slander me, I will still sue you for defamation. How could I, such an honest person, be in the limelight? Go ask the IMO training team, everyone knows that I, Qiao Yu, am the most low-key person!"

Chen Zhuoyang looked at Qiao Yu and said jealously, "Don't pretend. I was just discussing the graduation defense with my boss, and Professor Tan from the research center sent me a video of you giving a lecture to a room full of professors on the stage.

The boss was watching with great interest, and after watching it, he looked at me with a strange look in his eyes. But that wasn't the most uncomfortable thing for me. The boss also sent me the video of you showing off! Let me see how you did it! By the way, Junior Brother, that wasn't your presentation today, was it? "

A statement full of resentment.

It turns out that people are most afraid of comparison.

Qiao Yu could imagine the scene, where the students and teachers were respectful to each other and were discussing the upcoming graduation matters face to face.

You know, for a student of a big shot like Tian Yanzhen, this kind of opportunity is very rare.

It was probably because Chen Zhuoyang had been staying in the research center to deal with some trivial matters, and Qiao Yu also spoke up for him, that he got the opportunity.

If it were other students with average talents, this kind of thing would probably be finalized with just a few words on WeChat.

What should have been a beautiful memory in life was rewritten by a video. It is indeed a depressing thing.

But Qiao Yu felt that this had nothing to do with him. Senior Brother Chen should blame Professor Tan for sending the video to the tutor...

However, seeing Chen Zhuoyang's lost look, Qiao Yu couldn't say anything to shirk responsibility. After thinking for a while, he encouraged him straightforwardly: "Brother, don't you think that compared to me, you just lack confidence?"

Chen Zhuoyang said in a muffled voice: "Is it just lacking a little confidence?"

Qiao Yu said firmly: "Of course! You didn't attend today's lecture. In fact, if you went, you would have some ideas, and if you have ideas, you should speak them out loud. So what you really lack is just confidence. I don't have anything else, I'm just confident enough.

Anyway, I don’t know whether those ideas are right or wrong, but since the professor asked me, I dare to speak out. In fact, we are all ordinary people, but the more ordinary we are, the more confident we should be! Anyone who says that ordinary people don’t deserve to be confident is an idiot! "

Chen Zhuoyang shook his head and smiled: "Alright, Junior Brother. Stop calling yourself an ordinary person. Actually, it's not that I don't have confidence. I just haven't been in this place for too long and I really can't be confident.

But a senior has already said that my situation will be fine after I graduate and go out to work. As long as I stay away from non-humans like you and get along with other colleagues, I will find that I am still okay, and my confidence will gradually return. "

Qiao Yu nodded, and suddenly realized that his senior brother might be better at comforting people than he was.

However, seeing that Senior Brother Chen was in a normal mood, Qiao Yu was too lazy to talk to him any more, so he said, "Then Senior Brother, do you want to come in and sit down?"

Chen Zhuoyang shook his head and said, "Oh, no need. My tutor asked me to come here. He said that since he saw you were back, he asked you to go to his office. He has been waiting for you in the office today. I thought you would come back after dinner at Huaqing, but I didn't expect you to come back early. So I chatted with you for a few more words."

"I...ah, forget it...Brother, next time if you have something to say, can you just tell me directly? Then I'll go to Director Tian first! We can talk later!"

After saying that, Qiao Yu closed the door and rushed towards Director Tian's office in a hurry.

When a mentor calls you, you must be positive.

……

Five minutes later, Qiao Yu arrived at Director Tian's office door. As usual, he pushed the door open and asked openly, "Director Tian, ​​Senior Brother Chen just said you were looking for me?"

"Yeah. Just now, several professors from the research center sent me messages, all reporting to me about your great achievements today. How does it feel to make a big fuss at the Huaqing Conference Hall?" Tian Yanzhen looked up at Qiao Yu, joked, and then put down the phone in his hand.

Seeing this action, Qiao Yu strongly suspected that before he came, his tutor was still chatting with the professors who stayed in the Huaqing conference room.

Qiao Yu said rather embarrassedly: "Director Tian, ​​please don't embarrass me... You know me well. I really just suddenly thought of a very clever idea at that time, but I really didn't mean it!"

Tian Yanzhen put away his smile, his expression serious, and asked again: "No, Qiao Yu, I am asking you seriously, how does it feel to cause a big disturbance at the Huaqing Conference Hall?"

"Uh..." Qiao Yu looked at Director Tian's serious expression, thought for a moment, then nodded and said, "Well... I feel pretty good actually. Especially when there were so many professors in the conference room and no one raised any questions, I actually felt pretty good.

Oh, by the way, when Professor Everton asked me how to name the point I constructed, and saw that I didn’t say anything, he named it JOE point. It felt very different, and I felt secretly proud of it. "

Listening to Qiao Yu's serious tone with a bit of pride, showing off his mentality at the time, Tian Yanzhen smiled again and said gently: "Well, I hope you can always remember this feeling, and it would be best if you can get addicted to it. The process of mathematical research is often very boring, and more fun actually comes from the final results.

Especially in the future, you will encounter more difficult problems. You may be stuck somewhere and not get a solution for a long time. You have not experienced that kind of pain yet, but everyone who is truly determined to make achievements in mathematics may encounter it in the future, and it may even be a normal part of life.

Otherwise, why are there so many amazing mathematicians in history whose achievements have amazed the world, and why did they leave behind not only theorems but also conjectures that have troubled mathematicians for decades or even centuries?
You are very lucky to experience the joy of mathematical achievements at such a young age. Remember this feeling. If you really encounter a bottleneck in the future and feel that you can't hold on, think about how you felt today!

And you are at an age when you need to show off! Don’t care what others say. If young people don’t show off, should a bunch of old men be allowed to do so? As long as you have the ability, just show off. The teacher will back you up! "

Qiao Yu nodded in agreement, and then became even more proud.

Well, he was indeed feeling very happy today, but it was obviously not just because he had become a bit of a celebrity in the meeting. However, hearing his mentor say this made Qiao Yu feel even more proud.

Director Tian has said so, so if any of his fellow students feel jealous again in the future, he will have something to say.

After all, he was in a difficult situation, and was just pretending to be powerful. If you have any opinions, go and tell Director Tian. It's none of my business...

"By the way, Professor Ding was just saying that the professors at the conference were arguing. Some thought your idea was right, while others thought it was outlandish.

Professor Tan is very supportive of your idea. He said he thinks you are right. Try to write out the detailed proof process in the next two days. I know you have the ability to do it.

But don't feel pressured. Even if you are proven wrong in the end, it doesn't matter. Remember that the people participating in this meeting are all top experts in related fields. If your ideas can cause debate among them, it is a success in itself. "

Qiao Yu curled his lips and said, "Director Tian, ​​although I agree with most of what you said, I cannot agree with part of your last sentence!"

Tian Yanzhen was not annoyed by the rebuttal, and asked with a smile: "Oh? What part do you disagree with?"

Qiao Yu said confidently: "You said that all the participants in this meeting are top experts in related fields. Let me put it this way, I have explained the problem to them so clearly, but they still think I am fanciful. This is enough to prove that they are only half-baked and definitely not top experts!"

This one sentence made Tian Yanzhen silent.

He suddenly felt that calling this kid here was a waste of time.

This mentality is many times better than when he was young.

Good guy, he didn't even provide the complete proof process, and people's doubts were only half-hearted. This confidence is definitely higher than 99% of his peers. At least in terms of not blindly believing in authority, this guy has already stood at the top of the world pyramid.

But then again, Tian Yanzhen really likes Qiao Yu's strong character. And this kind of personality is really suitable for carrying the banner of Yanbei School in the future.

The only problem is that when this kid grows up and he steps down, if there are really disagreements in the future, he probably won't be able to influence this kid's decision. But it doesn't matter.

As long as Qiao Yu can win the Fields Medal before he turns 30, this guy will probably be the one to carry the banner of the Chinese mathematics community in the future. He will just have to wait and see. He hopes that Qiao Yu can give him a surprise and make the Chinese mathematics community have a stronger voice in the world mathematics community.

If Qiao Yu could do this and help him vent his anger, then all those trivial matters at home would really not be important.

"Okay, okay, okay, well said. There are still two days left for this seminar. If it doesn't work, we can extend it by one day. The last day is up to you. You can arrange your time by yourself. I was just hoping just now, but now it's a requirement. In short, you have to come up with something complete before the seminar. Let those professors who oppose you have nothing to say. Is that ok?"

Tian Yanzhen made the request straightforwardly.

This kid wasn't very polite to him, so there was no need for him to be too polite.

Qiao Yu patted his chest and said, "Don't worry, Director Tian. I have roughly figured it out. I just need to give an accurate definition of this local deformation model structure and then prove it.

You should also know that Professor Everton has directly defined the special singular point I conceived as a Joe point. Now we just need to complete the proof of the conjugacy of the Joe point. This is easy and can be done by tomorrow night at the latest. "

Tian Yanzhen glanced at Qiao Yu and said, "Okay, as long as you think there will be no accidents in the proof process, I will of course believe you. Once you have written the proof process, please contact me at any time. Send it to me first."

"No problem, Director Tian! I'll go first. I'll race against time so that you can see my proof process as soon as possible."

"Okay, go ahead. By the way, there will be a lecture by Professor Tan tomorrow morning, so you don't have to go. If necessary, I will ask him to give you a private lecture. You still have to go to Professor Phyllis's lecture in the afternoon. It's a rare opportunity."

"Got it, Director Tian."

……

Back in the room, Qiao Yu sat in front of the computer and opened LaTeX directly.

Next comes the proof.

The most important part of the paper is that the nonlinear conjugate ridge network structure holds, that is, the complete proof of the two distal singular points P1 and P2 on the algebraic variety, which have local ridge network singularities respectively and influence each other through nonlinear homological mapping.

On the way back, Qiao Yu had already figured out how to prove it.

The first step is naturally local structural analysis, which is nothing more than describing the geometric structure of the singular point by defining local equations, then checking the properties of the singular point by calculating the Jacobian matrix, and using blowing and analytical decomposition to study its ridge structure.

These are all ready-made methods, and Qiao Yu doesn't even need to think about them.

The key point is the construction of nonlinear homological mapping.

What really made Qiao Yu think was what tool to choose to calculate the local homology of P1 and P2. This was probably the only difficulty. Different homology theory tools directly affect the solvability when dealing with this part.

After careful consideration, Qiao Yu decided to use layer homology plus Grothendieck local homology to deal with this problem.

Sheaf homology can more conveniently capture the local geometry and topology information of algebraic varieties, while Grothendieck local homology provides deeper tools for processing local rings and algebraic structures, which can further analyze the properties of local algebraic rings of singular points and reveal the subtle algebraic structure of algebraic varieties at singular points.

This should be the simplest way to relate the local homology dimension of a singular point and the properties of a local ring through a homology mapping.

What Qiao Yu was pursuing was to use a simple and direct way to shut up those so-called senior professors who thought his reasoning was problematic.

In fact, we can also use layer homology plus De Rham homology, and Qiao Yu thinks that we can also reach the same conclusion. However, when dealing with analytic singularities or analytic forms on algebraic varieties, De Rham homology provides a differential geometry perspective, which makes the problem more complicated to solve.

There is no need to show off his skills in analytic geometry. Qiao Yu feels that his analytic geometry is not strong. If there is any loophole in the De Rham homology proof, both Director Tian and Master will be embarrassed...

After all, my tutor and my grandfather are both big names in analytic geometry.

In short, if this problem is solved, the whole proof process is almost complete. The next step is to follow the routine. As long as such a point exists, the functor derived by higher-order category theory will definitely fail.

The derived functors are not equivalent, so all the conclusions are naturally untenable.

The real difficulty lies in how to reconstruct the Ambidexterity theorem so that this key theorem can be valid again in the process of proving the geometric Langlands conjecture.

At this stage, Qiao Yu planned to solve this problem himself, but he would not tell the other party...

　Thanks to book friends 20230114003824260, book friend 20240919204135030, book friend 20241011205517969, and Yeji Xiaoshi for their rewards and encouragement!

　　The 36th day of the -word check-in is completed!
　　
　
(End of this chapter)