Chapter 103 Crazy Mathematics Rookie
Tian Yanzhen knew that Qiao Yu was immersed in the paper and did not want to disturb Qiao Yu, just like he did not want to be disturbed by anything or anyone else when he was reading the paper. .
Human thinking, especially mathematical thinking, needs coherence. When you are immersed in a certain state and are suddenly interrupted, it is difficult to get back to that state.
This is indeed the case.
Peter Schulz’s paper seemed to open a door to a new world for Qiao Yu.
Those complex algebraic symbols and high-dimensional geometric structures have never been displayed in Qiao Yu's mind as three-dimensionally as they are today.
He couldn't even describe this feeling in accurate words.
If Qiao Yu had to describe this feeling, it would probably be a counter-intuitive sense of geometry.
After reading this paper carefully, his brain seemed to be occupied by countless high-dimensional geometric spaces, but these spaces are not as smooth as the Euclidean spaces that can be seen everywhere in our daily lives. Not continuous, but divided.
Especially the rigid analytical space, in Qiao Yu's mind, it turned into a geometric object that was divided and layered infinitely. Each dividing line was so precise and subtle.
Qiao Yu couldn't explain why those lines appeared there, but his subconscious told him that these lines should appear there.
It is these complicated lines and geometric figures that make the space no longer continuous, but present a discrete but tight structure.
These structures undergo complex and complicated transformations in the P-advanced field.
There is no smoothness and intuition in this geometric world. What is there is just constant reconstruction by various expansion methods, infinite extension, and finally forming a universe.
A universe intertwined with algebraic symbols and geometric figures. Every point in the universe contains infinite divisions and infinite levels of details under a special law. These laws govern these points, lines, and surfaces, pushing the structure of the geometry of the universe to perfection...
The only regret is that one night is not enough for Qiao Yu to fully understand even the first paper.
When Qiao Yu woke up from his focused state due to a deep sense of exhaustion transmitted from his brain, it was already 12:23 in the night.
I really feel very tired, even more tired than before when he studied Lao Xue's Diophantine equation until two or three in the morning.
However, considering that he had been in the car for six hours today, Qiao Yu felt that it was probably normal to feel tired.
So on the first night after coming to the capital and being admitted to college, Qiao Yu climbed into bed without even washing up. What's even more annoying is that Qiao Yu, who has always slept well, had a strange dream that night. In the dream, he came to a strange maze world, which was composed of countless strange doors.
Every time Qiao Yu opens a door in his dream, he will see a magnificent space composed of various strange geometric configurations. This world is so wonderful, even the physical rules followed by a photon are completely different from the real world.
It might be possible to be here or there at the same time, so what’s the probability?
No, maybe it can be here, it can be there, it can be everywhere.
Until Qiao Yu opened a door again, he suddenly felt a burst of light shining into his eyes, and then subconsciously opened his eyes and woke up...
I forgot to close the curtains last night. The rising sun just hit his eyes through the window, and he felt dizzy before he even got up.
Qiao Yu quickly got up from the bed, picked up the mobile phone casually thrown beside the bed and took a look. Good guy, it was already 8:20.
He had never woken up so late when he was in Star City.
Even if I stayed up until one o'clock on the thirtieth day before going to bed, I got out of bed on time at seven forty the next day.
Qiao Yu was different from other children in this aspect since he was a child.
Other children can't wake up, but Qiao Yu only needs to sleep for more than seven hours continuously every day, and then take a break for ten or twenty minutes at noon, which will be enough for him to be energetic for the whole day.
Although he seemed to have had a dream all night yesterday, and he still clearly remembered those dreams, Qiao Yu felt that he was in good spirits and was not sleepy due to too many dreams, so he simply climbed out of bed.
Keeping in mind Lao Xue’s instructions yesterday, Qiao Yu put on her clothes and made herself neatly dressed before picking up the mouthwash cup and toothbrush from the window sill, and then put the towel on her shoulders before walking out of the room. Door, go to the second floor at the corner.
Lao Xue took him there yesterday. The restroom and washroom were both on the second floor.
Going up to the second floor, turning to another corner, Qiao Yu just walked into the bathroom, and a middle-aged man walked out of the bathroom. The two met face to face, and the other person opened his mouth and asked, "Huh?" Who are you? "
"Um, I am..." Qiao Yu was about to introduce himself, and suddenly he said: "Oh, you are the teacher's new student, right? Qiao, um, Qiao? What’s coming?”
"Qiao Yu, the metaphor of metaphor!" Qiao Yu bit the word metaphor very hard.
When Lao Xue asked him what his identity was yesterday, Qiao Yu didn't break his guard.
Today this middle-aged man didn’t even remember his name, which really made him lose his defense!
Sure enough, he is still a little Karami.
The middle-aged man said politely: "Ah, yes, yes, Qiao Yu! Hello, junior brother."
Qiao Yu felt a little emotional. As expected, his tutor was all over the world. This guy looked to be in his forties, and he still had to be called Teacher. His seniority was indeed very high.
Of course, Qiao Yu still asked politely: "Excuse me, who are you..."
"Oh, my name is Chen Zhuoyang, I am Director Tian's doctoral student, because I have to be responsible There are some lectures and things like that, so my office is upstairs, you can go up and sit there when you have time," Chen Zhuoyang said politely.
Although Qiao Yu looked very young and immature, the instructor specifically mentioned it to him, which proved that he valued Qiao Yu, so Chen Zhuoyang behaved very politely.
However, these words set off a storm in Qiao Yu's heart. Looking at the other party's almost half-bald head and high hairline, he couldn't help but hesitantly asked: "Well, Senior Brother Chen, can you take the liberty of Just a question, how old are you? ”
"Me? 29? What's wrong?" After saying that, Chen Zhuoyang looked at Qiao Yu's shocked eyes, subconsciously touched the top of his head, smiled to himself and said: "Haha, I haven't graduated yet, and my hair is about to fall out! People say I look like I'm in my thirties , Alas..."
"No, Senior Brother Chen, you are too modest. I believe you even if you say you are forty, really." Qiao Yu couldn't hold it back, and even forgot about Qiao Xirang. He reminded him not to cause trouble when he went out, and he said it very honestly.
"Well..." Chen Zhuoyang thought for a while and replied seriously: "Junior brother, you have to remember that it doesn't matter whether a man is handsome or not, the key is to be talented. And you haven't entered mathematics yet. I don’t know how difficult it is to learn mathematics well, but I’m pretty good at it.”
Actually, when Qiao Yu blurted out those words just now, she felt a little regretful and originally wanted to apologize.
But as soon as Chen Zhuoyang’s teaching came out, Qiao Yu couldn’t help it anymore: “Um, Senior Brother Chen, it’s really not me who is making excuses. But the author of the paper I read yesterday was already 24 years old. He is a top professor at a well-known university. You are already 29, and you haven’t even graduated with a PhD. His talent is incomparable. I looked at his photos online and he has a lot of hair.”
Chen Zhuoyang blinked and asked subconsciously: "Who are you talking about?"
Qiao Yu replied: "Peter Schultz."
Chen Zhuoyang stared at Qiao Yu silently until Qiao Yu felt a little scared in her heart, and then she said slowly: "Junior brother, thank you for comparing me with the youngest Fields Medal winner! But I still suggest you wait until you start. Come talk to me about this issue again on the day you start writing your master's thesis.
Although students who become teachers are not lacking in resources, you will know how strict the teacher's requirements are when the time comes. Senior brother, well... I have decided to treat you as nothing by then, and nothing you say will be of any use."
After saying that, Chen Zhuoyang raised his head proudly and left.
The sound of "dong dong dong" going up the stairs reached his ears. Qiao Yu didn't even react yet. He was probably angry, right?
Senior Brother Chen is so cute. He doesn’t talk trash when he is angry. Even threats are cute and have no deterrent effect.
The only pity is that we left too fast.
He also wanted to tell this senior brother that if he couldn't even complete his master's thesis, he would definitely choose to drop out of school.
Are you kidding, a master's thesis, isn't that something you can write if you have the brain? No matter how strict Director Tian is, he won't ask him to solve a world-class mathematical problem for his master's thesis, right?
Muttering these messy things in his heart, Qiao Yu quickly walked into the bathroom, cleaned himself up, returned to the room, put the things back in their place, and then took the meal card and went out to have breakfast.
The research center is some distance away from the canteen where Lao Xue took him to yesterday, but Qiao Yu thinks this is actually quite good. Just walking back and forth can be regarded as a morning exercise. And it's the winter vacation, so there aren't many open canteens. It should be more convenient after school starts.
I went to the canteen and bought two big buns and a cup of soy milk, and finished them on the way back.
Then Qiao Yu opened yesterday's paper again.
I don’t know if it’s because my mind is clearer in the morning. Yesterday, there were some proof processes that I couldn’t understand, but today I felt that I understood them after reading them again.
We can now roughly understand why some Chinese articles refer to Peter Schulz’s pioneering research as a quasi-complete space. Because this theory can really be described as mysterious.
For example, when it comes to etale homology, the simplified calculation framework provided by the complete space. Yesterday I couldn't figure out how to do this at all. After reading it today, I found that it is nothing more than the homology class in some complete spaces being mapped to the etale homology class of classic geometric objects through the structure of the complete space.
In this way, the homology class structure of algebraic varieties can be calculated more efficiently in the p-adic background.
After feeling that he thoroughly understood these abstract things, Qiao Yu even wrote a related mathematical question on his own paper based on what he couldn't understand yesterday.
After finishing the question, Qiao Yu thought it was cool, so he picked up the pen again and answered the question based on his understanding of the theorems and lemmas given by Peter Schulz in the paper.
After finishing, Qiao Yu instantly felt cute. He looked at the time and saw that it was already half past ten in the morning. So he stood up and was about to go out to exercise. When he turned around, he saw Director Tian and Lao Xue through the window. They were walking towards the room together.
Coincidentally, the two of them were looking at him through the window, so Qiao Yu immediately reacted, ran to the door and opened it.
"Director Tian, Teacher Xue, you are here, please come in."
"Well, I came here just to see you. How is it? Are you still used to it?" Tian Yanzhen came in. Shi asked.
"It's okay, I'm quite used to it. It's just that the water for washing my face in the morning is a bit icey, but everything else is fine." Qiao Yu said truthfully.
"Don't you know how to buy a kettle and get some boiling water? This kid..." Lao Xue was speechless at Qiao Yu's complaints.
"Oh!" Qiao Yu responded honestly.
"Okay, it's normal that you haven't gotten used to it when you first came here. You're so young, it doesn't matter if you just wash your face with cold water, you'll be more energetic." Tian Yanzhen said that he had already walked to Qiao Yu's table and took a look. He threw the manuscript paper casually next to the computer, and then picked it up with interest.
"Um, Director Tian, this is a random question I asked after reading a paper yesterday." Qiao Yu explained quickly.
This proves that he is very meritorious. He started studying as soon as he arrived at Yanbei University.
"Yes, I know." Tian Yanzhen responded casually, and then carefully looked at the questions and the solution process on the manuscript paper.
Xue Song on the other side glanced at the paper on the computer, and then turned to look at Qiao Yu with a confused expression: "What paper are you reading?"
Qiao Yu answered. Said: "The paper Peter Schulz published in 2011 on the P-adic Hodge theory of rigid analytic spaces and complete spaces."
Xue Song cried He couldn't help but look at Qiao Yu and said, "Didn't I tell you yesterday to download Professor Robert Greene's paper and study it? Are you going to listen to the professor's lecture tomorrow without any preparation? Look at Peter... What's Schulz's paper for?"
"Huh?" Qiao Yu then remembered that Lao Xue had indeed mentioned it after setting up the account for the library software yesterday.
After smiling sheepishly, he explained: "I forgot. Didn't you tell me about the deeds of Zheng Zhiqiang and Peter Schultz yesterday before leaving? Zheng Zhiqiang was a computer engineer, so I didn't. Then I downloaded Peter Schulz’s paper and became fascinated by it.”
This answer left Xue Song speechless.
He told Qiao Yu yesterday that these two people seemed to be telling him not to be proud, but in the end, this guy went to study the complete space theory pioneered by Peter Schultz?
This is not a typical case of not learning yet...
Xue Song was about to teach Qiao Yu a few words when Tian Yanzhen suddenly handed Qiao Yu's manuscript paper to Xue Song and said, "Professor Xue , take a look too.”
"Uh, okay." After taking the manuscript paper from Tian Yanzhen and looking at the title, Xue Song didn't know what to say.
Good guy...do you really understand?
On the other side, Tian Yanzhen was already chatting with Qiao Yu: "Have I ever told you that the mathematical knowledge you are currently exposed to is fragmented?"
"Yes. "Qiao Yu nodded honestly and replied blankly: "Last time when I was at CMO, You told me in the dormitory that time.”
“Well, so I worked with Professor Xue to help you develop a study plan, preparing you to make full use of this half year to master the entire algebraic geometry. A rough overview of the number theory system
Professor Xue also helped you develop a complete set of learning and training courses, but I suddenly changed my mind just now. During this half year, you can study on your own by reading papers in the library's database based on your interests.
If you don’t understand anything, just ask Professor Xue or me. Professor Xue should be able to spare time to answer your questions at any time. I will spare half a day every week to discuss related issues with you.
If there is something beyond the research scope of Professor Xue and I, there are many professors in the research center, you can go and ask for advice. I will ask Professor Xue to send you a catalog later. It contains the contact information and research directions of all the professors in the research center and the School of Mathematics.
I will say hello to you then, but when you ask other professors for advice, you must be more humble and respectful. Do you understand? Everyone will answer your questions for free. "
Qiao Yu immediately nodded and assured: "Don't worry, Director Tian, I understand this. To be honest with you, I am actually the most polite. ”
Tian Yanzhen explained again: "Just remember, you'd better not read Schulz's papers this afternoon. Download two of Professor Robert's papers in the past five years and take a look. You should have been exposed to some of them. It can help you understand what he will probably talk about tomorrow.
This will make you cherish attending this kind of lecture. Taking the opportunity to attend is also the most basic respect for the specially invited lecture professor, and it is also the most basic courtesy. Not only this time, but also other lectures that interest you in the future. ”
"Understood, I will definitely remember this time, and I promise to read that professor's paper in the afternoon." Qiao Yu said immediately.
While the two were talking, Xue Song had already gone over Qiao Yu's question and the solution process.
The mood is very complicated.
He had also read this early paper by Peter Schulz.
He was still at Princeton at that time.
To be honest, this thing was obviously too high-end for Xue Song at the time. Well, actually, the same is true for him now.
A complete space is a geometric object defined on a p-radical body. The p-adic number itself is a relatively abstract and complex mathematical system, which is completely different from the intuitive real number or complex number system.
What’s even more frightening is that this theory also combines multiple branches such as algebraic geometry and topology.
This is no longer a simple study of geometric objects, but also requires the ability to understand the algebraic structure of geometric objects and their behavior in different number fields.
If the above difficulties can be overcome, then the tilt theory involved in complete space will really make people scratch their heads. This idea relates a p-adic perfect body to the perfect body of characteristic p by introducing a new algebraic geometry perspective.
But this connection is very abstract and involves extremely abstract mathematical techniques such as algebraic closure and integral closure...
In other words, this theory cannot be understood with the help of existing mathematical frameworks. , to understand Peter Schulz's ideas requires building a completely new mathematical system in the mind.
Qiao Yu just downloaded a paper on this subject yesterday and took a look at it. It seems that he can understand it today? !
Is this the advantage of a poor foundation? No, if the foundation is not good, you should not be able to understand this kind of paper at all.
To be honest, Xue Song found it difficult to understand how Qiao Yu understood the concepts created by Peter Schulz.
Xue Song actually heard what Tian Yanzhen said just now, but he couldn't raise any objections. He just silently put Qiao Yu's manuscript paper back on the table.
This is probably a stocking strategy. But the resources prepared for Qiao Yu are all top-notch. Just watching him tinker with it every day and see what results he can produce seems to be something very exciting.
"Professor Xue, what do you think?" Tian Yanzhen asked. Xue Song shook his head, showed a wry smile, and said: "I think this arrangement is good. Anyway, it will only last half a year. Let him have enough freedom first, and then let the results speak."
Tian Yanzhen nodded, looked at Qiao Yu and said: "That's it, do you have any questions?"
Qiao Yu quickly asked: "By the way, Well, is there another Senior Brother Chen in this building?”
Tian Yanzhen nodded and replied: "Yes, his office is arranged on the third floor, have you met?"
Qiao Yu kept nodding and said: "Yes, yes, yes, I just want to What is the topic of Senior Brother Chen’s doctoral research?”
Tian Yanzhen looked Qiao Yu up and down, and was amused: "Why, are you addicted to giving advice to your fellow students? You used to help Professor Xue's master's students revise their papers and feel that it does not reflect your level. Do you want to give advice to the doctor?"
Qiao Yu denied it repeatedly: "No, no, I just want to hear how difficult it is for Senior Brother Chen to choose a topic, so that I can be mentally prepared for future Ph.D. topics."
Tian Yanzhen shook his head and said: "Your situation Unlike Senior Brother Chen, his topic selection has no influence on you. "What a reference."
However, after finishing speaking, seeing Qiao Yu's eyes full of curiosity, he still revealed: "His research direction is geometric analysis on complex manifolds, specifically extremes. The existence of small surfaces on Keller manifolds Properties and stability.
For example, how to construct a minimal surface through the variational method, the stability of the energy functional of the minimal surface, and the evolution of the minimal surface under the action of Ricci flow, and so on. ”
"Oh!" Qiao Yu nodded, looking stunned.
"What? Have you studied these?" Tian Yanzhen asked.
"No!" Qiao Yu shook his head and said: "I just think there is a reason why my senior brother has not graduated at the age of 29. When I watched the video before, a big boss said that it involves functional Everything analyzed is quite difficult.”
Xue Song couldn’t help it and said, “Functional analysis is difficult? After fully understanding Hilbert space and Banach space, learning functional analysis is very simple."
Qiao Yu looked at Xue Song blankly and replied: "Is that what I am looking for? The p-adic number analysis method involved in analyzing infinite-dimensional space in the paper is to use the techniques in functional analysis? P-adic Banach space is used in many places in the paper. ”
Tian Yanzhen nodded and replied: "When doing analysis related to the topological properties of p-adic numbers, function spaces and p-adic representation theory, the structure of Banach space does often appear. However, P-adic numbers are special and involve Banach space is different from real number analysis, that is, what you said is that P enters Banach space.
Don't worry about this. You will understand it when you have a broad knowledge. At the forefront of mathematics. In research, Many mathematical tools are intertwined. This is why mathematicians need to have a comprehensive foundation. But I want to see how far you can go in your own way. "
Qiao Yu? He immediately shook his head and said, "No more."
"Then let's go first. Professor Xue will come to take you to Professor Robert's lecture tomorrow." After saying that, Tian Yanzhen followed Xue Song Leave together.
Qiao Yu watched the two professors leave, stood up and moved his body, and read the paper again.
Director Tian said, just start reading Professor Robert’s papers in the afternoon.
It is now more than an hour before eating, and Qiao Yu feels that he should have almost absorbed the nutrients in Peter Schulz's first paper.
......
The two mathematics tutors walked silently for a while under the warm sunshine. I think there's nothing wrong with it, but how should we arrange it for him after school starts next year or let him go to the elite class? Study together?"
Tian Yanzhen thought for a while and said, "Don't make a decision in a hurry. Let's see how far this child can go first. To be honest, I don't know what to do now. Teach him, I just think it might be more appropriate to let him take the initiative to explore the problem himself.”
Xue Song nodded. To be honest, he had no good way to teach Qiao Yu. Even when he saw the manuscript paper where Qiao Yu asked and answered questions, he began to doubt whether he had the ability to be Qiao Yu's little guide. .
It is no exaggeration to say that there are not many mathematicians in the world who can understand the content of Peter Schulz’s research. After all, this is the most basic research in mathematics, aiming to build a bridge between algebraic geometry, number theory and advanced analysis fields.
Qiao Yu seems to have an extremely amazing talent in comprehending these complex mathematical ideas. Not only has he never taught such a student, he has never met him before, and he suddenly felt a lot of pressure.
“Don’t think so much. Speaking of Peter Schulz, there is another interesting thing. He gave his thesis to his supervisor Lambert. After reading it, Lambert told him that he could graduate with a Ph.D. "So true geniuses don't need us to worry too much."
Tian Yanzhen added optimistically.
After hearing these words, Xue Song took a deep breath and couldn't help but ask: "When you were teaching at Princeton, you came into contact with many students. Have you ever met anyone with Qiao Yu's talent?"
Tian Yanzhen smiled and said : "You also studied at Princeton School of Mathematics for eight years. You should know better than me what the situation of your classmates is like, right?"
Xue Song shook his head and replied: "There are really many geniuses. Among them are the very common ones , but there is really no such thing as a genius that I admire from the bottom of my heart."
"That's because you are also one of the geniuses." Tian Yanzhen said with emotion: "You can. Students who successfully graduated from Princeton, For ordinary people, they are all geniuses, let alone those who can graduate with a Ph.D., but geniuses in the fields of mathematics and theoretical physics are divided into three levels. "
In a word, let Xue! Song Song completely lost interest in chatting.
It’s really a hopeless field. Geniuses are divided into three, six, and nine grades...
"If Qiao Yu is really the kind of genius I think he is, I have to thank you. If it weren’t for Na Tong If I miss the call, I'm afraid I will regret it for the rest of my life." Tian Yanzhen looked at Xue Song and said sincerely.
"You are serious!" Xue Song said politely.
"Okay, I'll go back first, and you can do your work. Oh, by the way, the joint training plan with Yu University has been drawn up, and I have helped your students gain some rights. If their results during the joint training period meet the standards of Yanbei University, they can choose to get a diploma from Yujiang University or Yanbei University.”
“Oh, thank you so much!”
"Xiao Xue, you're welcome!"
Seeing Tian Yanzhen walk into the small building next to him, Xue Song stood there and thought for a moment, then smiled and took out his mobile phone and walked out of the research center. While editing the message.
This news can probably give his doctoral students some blood, right? !
......
At noon, Qiao Yu went to the cafeteria alone for lunch. After coming back and taking a nap for ten minutes, Qiao Yu began to search for Robert in the backstage of Yanbei University Library. The professor's paper was downloaded.
Listening to what needs to be heard is also an essential advantage for students.
Especially what the instructor has repeatedly emphasized, and even related it to politeness and respect, that is what you must listen to.
As for the others... you actually have a choice.
People who can become important people will most likely not have to worry about everything with students. Anyway, that's how Qiao Yu understood it.
Just like Principal Zhang of Xingtie No. 1 Middle School.
As long as you follow his requirements and get results, Lao Zhang is really tolerant in other aspects.
Qiao Yu felt that even if he was so bored that he took down a few of Tieyizhong's signs, Lao Zhang would smile and ask the school's logistics department to make new ones, and then tell him that he would not do it again.
I searched for Robert Green’s name in the paper retrieval system of Yanbei University Library, and a bunch of papers suddenly appeared.
Qiao Yu was shocked. But I soon discovered that not all of them belong to the same person.
It seems that there are many people named Robert Green abroad.
Although I encountered a similar problem when searching for Peter Schulz, there was only one interference term, and the guy studied chemistry. The direction of the thesis is completely different.
But Robert, this guy, has many papers in mathematics.
Fortunately, Qiao Yu found that this paper retrieval system is actually very easy to use. It is not only rich in content, but also allows you to choose the age. The advanced search page even supports searches for the author's affiliation.
Qiao Yu remembered that Lao Xue said that this professor was from New York University, which was much more convenient.
Soon, the serious Professor Robert's paper was downloaded.
I don’t know if it was because he studied Peter Schulz’s paper first that Qiao Yu’s brain opened up again. Qiao Yu actually felt that it was quite easy to understand the professor’s paper.
Well, it seems a bit easy to say it is easy, but at least it is not difficult.
For example, Qiao Yu really feels that those lemmas, preconditions of theorems, a series of concepts, and the proof process are all easy to understand. It doesn’t take too many brain cells to understand. But this kind of work-rest balance is pretty good.
Reading Peter Schulz's paper yesterday was indeed too hard on your brain. Today, you should relax while reading a paper that is not so difficult to understand.
Although he was relaxing, it was already nine o'clock in the evening when Qiao Yu finished reading the two papers, and went to have dinner in the middle.
After putting down the paper, Qiao Yu began to think habitually again, and suddenly an idea came to his mind.
The content of Professor Robert's research is simply the problem of accurately estimating the upper bound of the number of rational points for a given type of algebraic curve, especially a high-dimensional algebraic curve. This type of problem is actually closely related to the Diophantine equation. .
Find the number of rational points and then study the distribution of these rational points.
It is simply that the geometric structure of high-dimensional algebraic varieties is often more complex, with more complex singularities, topological properties and different homology properties. These geometric properties all affect the distribution of rational points.
So there is actually only one research goal for this type of problem, which is to simplify the process of finding rational number points as much as possible, and to easily find the distribution of rational number points. It is equivalent to given a high-order Diophantine equation, which can quickly determine whether there is a solution and solve this type of equation.
Well, in short, this is how Qiao Yu understands it.
This is the understanding of a layman in mathematics. If Lao Xue were here at this time, after listening to Qiao Yu's thoughts, he would probably want to beat up this guy who doesn't know how high the sky is.
The reason is also very simple, the research goal is simply ridiculous.
Simplify the process of finding rational points, but it is almost impossible to easily find the distribution of rational points on high-dimensional algebraic varieties. This is common sense in mathematics. What everyone is doing now is nothing more than efficiently estimating the number of rational points through geometric and algebraic tools, and understanding their distribution through modern algebraic geometry tools.
As for solving the Diophantine equation quickly?
Even if it is determined that there is a solution to the solution of elliptic curves or more complex equations related to modular form, if he really wants to solve it, Lao Xue can only say haha.
Of course, these are not problems for Qiao Yu, a layman who doesn’t have much respect for mathematics. In addition, he just studied Peter Schulz’s mathematical ideas yesterday, which is a very bold idea. The idea suddenly popped up in Qiao Yu's mind and was out of control.
Why can't he try to use the theory created by Peter Schulz to solve this type of problem?
Regardless of whether it works or not, he can try to introduce a complete space into it. There is no suitable tool to deal with similar problems, but he can also create it himself.
Although this is a framework built by others, as long as it is within this framework and conforms to the rules of this framework, it is definitely feasible to create tools as long as it can solve the problem.
So the question before Qiao Yu now is very simple. How to introduce the problem of upper bound estimation of rational points with algebraic curves into the framework of quasi-complete space theory?
Qiao Yu, a newborn calf who is not afraid of tigers, sat at the table and fell into deep thought.
A pen also began to scribble on the manuscript paper.
Okay...
This problem seems not that simple, the main problem is the transformation of the problem.
After thinking for a long time, Qiao Yu came to the conclusion that if the upper bound estimation of rational number points can be transformed into a problem of homology and geometric properties on a complete geometric object, then it is logical to use the deep tools of p-adic geometry, such as complete Algebraic spaces, geometrization of modular forms, and p-adic homology theory are used to analyze these rational number points.
I just don’t know if this transformation will make the problem more abstract and complex.
But it doesn't matter, he is just a little Karami anyway, he is just playing. Is it free to try it?
So soon Qiao Yu wrote this paragraph on the manuscript paper with great interest:
“Suppose X is a high-dimensional algebraic curve defined on the number field K, and X is a closed subset in p-adic complete algebraic space, then there is a constant C that depends on the geometric properties of the curve X, such that the number of rational points on the curve satisfies: N(X)≤C ”
Naturally, N(X) represents the number of rational points on the curve X.
It was just the intuition that came out of Qiao Yu’s brain just now, that there must be such a constant C. The reason is very complicated, and it is related to the geometric configuration of the curve in the complete space. You need to have an understanding of Peter Schulz's theory to understand this proposition.
Now the first step he needs to do is to prove this proposition.
Because as long as the existence of this constant C is proved, this conclusion will provide a solid theoretical basis for estimating the upper bound of the number of rational points on complex high-dimensional algebraic curves.
After proving the first step, it is to find the formula of the constant C and prove that this formula is correct.
Then - problem solved!
However, when Qiao Yu was full of ambitions to prove this proposition, he suddenly felt that the question he raised seemed a bit impossible to start with.
He seems to be stuck in the cycle of steps it takes to put an elephant in the refrigerator.
The first step is to open the refrigerator door. The second step is to put the elephant in. The third step is to close the refrigerator door.
The only problem is, it seems he hasn’t found a refrigerator as big as an elephant yet!
In particular, Qiao Yu suddenly discovered that even if this constant C formula really existed, it would not only depend on the geometric properties of the curve, but also may depend on the characteristics of the number field K, the modular structure of the curve, and even other algebraic geometry tools.
Because after racking his brains, Qiao Yu found that the existing algebraic geometry tools did not seem to support finding this C.
A normal math person would probably give up at this time, but Qiao Yu was different. He was just a math rookie, and he had already regarded this challenge as a game.
Although I have no clue, what if it succeeds?
And again, if you don’t have tools, you can make it yourself.
It makes no sense that Peter Schulz could create such a powerful theoretical framework when he was only 21 years old. It makes no sense that he couldn't create a few useful mathematical tools at the age of fifteen, let alone the entire theoretical framework. They are all provided by others, and he only needs to create them twice within the framework, which is significantly less difficult.
After all, the rules are already there. He only needs to prove that his tool is correct through rigorous mathematical logic within the constraints of this framework rule.
So the next work can be further simplified. What kind of algebraic geometry tools can help him prove the existence of this constant C.
Qiao Yu thought for a long time with a frown on his face, and then confirmed again that first he needed a new homology category tool.
So another row of writing appeared on the manuscript paper:
"The homology category QH(Cp) is an enhanced homology category, defined on the completion space of the algebraic curve Cp. Its basic object is the traditional homology class H^i(Cp, Zp), but we need to perform special processing on it, Through a new operator Q, which acts on the homology class, the homology class Each object in the homology category not only has a topological structure, but also has an additional invariant..."
Huh... Qiao Yu looked at this expression with satisfaction. With this new homology category, It can decompose the homology group of the curve more precisely and greatly simplify the steps of proving the constant C. Perfect!
Sure enough, studying mathematics makes people happy!
Now a new question comes again, how to define this new operator Q, Qiao Yu feels stuck again...
MMPD, forget it! I can’t figure it out, so let’s put this aside. Anyway, to prove the constant C, this tool is not enough...
So Qiao Yu, who had gone completely crazy, started to create a second tool. Now he needed a new fuzzy measure function to approximate the constant C.
"Algebraic curve P-adic fuzzy measure μfuzzy(Cp) is a new measure function used to describe the fuzzy properties of algebraic curve Cp in a p-adic geometric environment. Its definition is as follows..."< br>
The 18th day of the 10,000-word update has been completed!
Thanks to book friend 20201229074741818, book friend 20241005192534569, and Rainbow x for their rewards and encouragement!
Another: After reading the book review section, I found that there are actually math book friends who are reading this book. I would like to emphasize again that all the so-called new mathematical theories in the book are all made up by the author. There is no such thing as any There is no reference meaning, and there is no mathematical logic at all!
This is just a novel. Brothers, just read it and have fun. The author is really crazy! If someone does develop similar new mathematical tools or theories, it is purely a coincidence!
(End of this chapter)