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Chapter 111 Thesis Completion
In Beijing, after putting down the phone, Qiao Yu excitedly cast his eyes on the formula on the computer, and began to re-examine the geometric background of the parameters and constants in the formula.

Qiao Xi was right. There were many parameters and the formula was complicated. He now needed to find the commonality of these parameters. So what was the commonality behind the conditions? No, it was not just about commonality, but essential commonality!
Otherwise it is not enough to tie these parameters together.

So naturally, Qiao Yu proposed a hypothesis: whether it is modular form, -adic geometry or quantized homology category, their parameters can be uniformly represented by a single geometric quantity.

The most critical part of this hypothesis is to find a unified geometric quantity that can capture the parameters reflecting the complexity of the curve in different geometric tools.

The next step is the most troublesome and most critical step.

Qiao Yu began to analyze the core parameters of each geometric tool one by one, then listed the core parts, and then found the common points of all these parameters.

To put it simply, whether these parameters are controlled by a common constraint parameter, of course, this is only the first step.

After finding the common constraints, we have to find the connection between these parameters and the common constraints. This is another extremely complex proposition. But how should I put it? It completely meets Professor Zhang’s requirements, complicating the proof process and simplifying the results.

It can also make this formula a truly universal formula. When solving the upper bound problem of rational points on a curve, you don’t need to think about anything else and can directly use his Qiao Yu-Qiao Xi Theorem.

Immersed in the proof of the theorem, the sky darkened before I knew it.

It was not until there was a knock on the door that Qiao Yu was awakened from his tedious work.

He looked out the window, then Qiao Yu stretched lazily on the spot and went to open the door.

Brother Chen was standing outside the door. When he saw Qiao Yu standing in front of the door, he immediately raised the lunch box he had just bought from the cafeteria and said, "When I went to eat just now, I saw that you were quiet in the room. I didn't want to disturb you, so I didn't call out. I thought you must not have eaten yet, so I brought you one."

"Thank you, Brother Chen!" Qiao Yu grinned and gave his brother a sincere smile, then took the lunch box, turned around and walked into the room.

After wasting an entire afternoon of brain cells, Qiao Yu did feel very hungry.

This lunch box arrived very timely.

Chen Zhuoyang followed in, muttering, "Don't be so lazy when studying that you forget to eat and sleep. You have to eat no matter how busy you are. No matter what you do, your health is the most important thing."

"I know, Brother Chen. Today is a special case!" Qiao Yu explained with a smile, then opened the lunch box and started to eat it voraciously.

Although I didn't do much exercise in the afternoon, I used up too much brain energy and still felt very hungry.

Even hungrier than after playing basketball all afternoon.

Chen Zhuoyang took a curious look at the contents on Qiao Yu's computer. Wow, there were those complicated formulas and all kinds of messy custom symbols again, which made his senior sigh inwardly.

There is no comparison and no way to compare.

"Are you still working on your topic?" Chen Zhuoyang asked casually, pointing at the computer.

Qiao Yu looked up at the monitor, swallowed the food in his mouth, and then said, "Yeah, it's a pity that there is no result yet."

In the past, he would have definitely boasted about it, but now that he had just found a clue and realized the difficulty of this problem, Qiao Yu became much more cautious.

He didn't want to brag before he produced results.

Chen Zhuoyang said self-pityingly, "Don't be impatient. How can results be so easy to come by? My boss introduced me to a research group before. Our main focus is on the geometric structure of high-dimensional algebraic varieties and the quantization of the moduli space. Our main goal is to publish a paper exploring the relationship between modular forms and Kähler manifolds... Oh..."

"Eh? Why are you sighing? Didn't get your paper published?" Qiao Yu asked curiously.

"Don't even mention it. We worked on it for more than a year, but a team at Harvard who did the same work published several papers that were better than ours, and then the project was hastily terminated. The paper was finally published in a random second-tier journal. The key point is that I was only the third author."

Chen Zhuoyang said sadly.

Qiao Yu probably understood Brother Chen's pain. He followed a top mentor, who also provided him with resources, but he failed to grasp them...

Of course, this cannot be entirely blamed on Senior Brother Chen. After all, a research group can be considered a whole.

But in other words, if Senior Brother Chen joined in and was able to turn the tide, the ending would probably be rewritten.

"Oh, Brother Chen, what was your main job at that time?" Not knowing how to comfort him, Qiao Yu simply asked.

"Classification of modular forms of algebraic curves, and research on their relationship with K stability. Well, let's not talk about this. Actually, I want to tell you that the first draft of my thesis is almost finished. When are you going to Huaqing? I can take advantage of your next visit to finish the final part." Chen Zhuoyang said his true purpose hesitantly.

Or you're too thin-skinned.

Qiao Yu felt that if it were him, after having done so many things before, he would definitely not say so much nonsense at this time and only reveal the real purpose at this time.

But he thought about it and said, "Hey, Brother Chen, what are you talking about? It's such an important matter, I can go to Huaqing for you! But Master said the other day that he is busy this week and may not be free until next week. If you want Master to help you personally, just give it to me before next Wednesday."

This time point actually has nothing to do with when Mr. Yuan is available. Qiao Yu simply feels that if his direction is correct, the results should be ready before next Wednesday.

At that time, it would be more convenient whether he helps his senior brother to look at the paper first, or really takes it to Huaqing to ask his senior master to help him take a look.

He was in no mood to care about anything else until he was sure that this direction was correct.

Chen Zhuoyang said happily: "Next Wednesday? No problem! By the way, I don't ask Mr. Yuan to really help me revise the paper. You just need to ask Mr. Yuan to help me find where the problems are and give me some specific comments."

Qiao Yu nodded and said affirmatively: "OK! Just remember to give me the paper before Wednesday."

"Thank you so much, Junior Brother. I'll go back and work on my thesis now."

"Well, it's okay, just go and do your thing!"

"Oh..." After watching Chen Zhuoyang leave, Qiao Yu sighed and suddenly realized that he now had more and more things to do.

Study, read, coax my supervisor and his supervisor, do projects, write papers, participate in selections, and then go to IMO to win medals, and by the way, beat up friends of the same age... Now I also have to worry about my senior's doctoral thesis.

How many things does he have to do alone? This is probably the legendary saying that those who are capable should do more work! Well, it seems that he must take on the task of revitalizing the Chinese mathematics community! After all, he is now sixteen years old and is no longer the fifteen-year-old kid!
Thinking of this, Qiao Yu, who had just eaten, became excited again. He sat upright in front of the computer and worked... In order to revitalize Chinese mathematics and to teach his tutor, master's uncle, and senior brothers a lesson, he had to work out Qiao's upper bound theorem no matter what!
……

Mathematical research often has an interesting feature, and it is a situation that countless mathematicians have encountered. That is, in the research process, it is very likely that you will get stuck at a certain step or a certain problem and make no progress for a long time.

Yes, it's stuck there alive.

Sometimes, after a sudden realization, you have overcome a hurdle and feel suddenly enlightened, with a smooth and bright road ahead.

But unfortunately, for most mathematicians in the world, this hurdle may take a lifetime to overcome, so the project ends in vain, and the previous work and data are sealed there. They fantasize that one day they will suddenly have an epiphany and let these studies see the light of day again in the future, but the greater possibility is that there will be no future.

The same is true for Qiao Yu, except that his talent is slightly higher than that of countless ordinary mathematicians.

When he realized the need to find commonalities in parameters under Qiao Xi's suggestion, this problem no longer seemed to be a problem for him.

All the previous reasoning and proof processes have been completed. They can be simplified by finding commonalities. The commonalities are hidden in the less obvious connections behind those parameters. As long as the work is detailed enough, Qiao Yu feels that this is definitely the right direction!
This is true.

For three days, Qiao Yu hardly went out of the house except for eating, and he didn't even read books. He devoted himself to this work wholeheartedly, and then he really discovered the existence of commonalities.

The higher the modular form level, the more complex the curve, so k-curve complexity.

The prime number p controls the local geometric behavior of the curve in the -adic number field. Different prime numbers correspond to different geometric constraints. The prime number p is also related to the complexity of the curve, so p is the local geometric complexity.

The parameter q in quantized homology reflects the influence of quantized geometric objects on the global complexity of the curve, which is a further quantification of the geometric complexity of the curve, so q is the global geometric complexity.

In other words, although different geometric parameters come from different sources, they all reflect the complexity of the curve from different perspectives.

What is this? This is the definition of parameter unification.

So on Friday night, Qiao Yu designed a unified geometric constraint parameter θ and proposed a second hypothesis: the geometric constraint parameter θ is a weighted combination of modular form rank, -adic number field primes and quantized homology parameters, which together reflect the global complexity of the curve.

Based on this assumption, it is obvious that we can get a basic structure: θ=f(g, k, p, q).

Of course, at this point, it is clearly not enough.

Because the weight of each parameter is not the same, in order to make the structure mathematically reasonable, a combination that can perfectly reflect the weight of each parameter is needed.

The next step is calculation and verification, which is complicated but not difficult.

But in just one night, he concluded that the growth of k grows logarithmically with the genus g, so: kglog(g); the complexity of local geometry changes exponentially with the increase of genus, so pe^g/2; in quantized homology, the growth of the relationship between the parameter q and the genus g directly calculates an approximate value: qg^3/2.

The formula comes out naturally: θ=f(g, k, p, q)=glog(k)+g^2log(p)+gq
After directly substituting the expression of the three parameters into the equation, we get: θ=glog(glog(g))+g^2log(e^g/2)+gg^3/2
At this point, the only important parameter left is the genus g.

The next step is the simplest simplification: θ = g(log(g)+log(log(g)))+g3/2+g^5/2
After three days of working day and night in front of the computer, Qiao Yu finally typed out the final formula for estimating the rational number points of the curve on the computer at 2025:2 p.m. on Friday, February 21, 11: N(X)≤C(θ)=θ^g
θ is the geometric constraint parameter he designed, and g is the genus.

This formula... is really beautiful!
After admiring it for a while, Qiao Yu immediately began to verify it. After all, a formula is useless if it only looks beautiful. It must be useful.

What he had to do was to determine whether his formula was accurate.

Qiao Yu chose the classic elliptic curve y^2=x^3+x
According to the known conditions of the BSD conjecture, we can know that the curve genus is 1. Substituting it directly into the formula and then simplifying it, the result is: θ=5. Well, 5 to the power of 1 is still 5.

The conclusion is obviously correct.

Because this is the classic Hermite curve, and the rational number points on the curve had been calculated more than ten years ago.

Next came the Model curve, special cases of the Fermat curve, various cases of the Kubert curve... Qiao Yu tried them all.

For example, the Model curve: y^2 = x^3 + k, where k is an integer. He verified the cases of k=-1, k=2 and other known finite rational points, and the results were all correct.

Then Qiao Yu opened Professor Robert Greene's paper and used his own formula to make comparative calculations with the formula derived by Robert Glenn. In terms of certain points, his formula was mostly the same as Robert's results, but for some uncertainties, there were some differences in the calculations of both parties, but not large.

Well, I don’t care who is right or wrong.

At least at this point, he can start writing the paper, and this step is actually the easiest for him.

Because he had already written down the process of deriving the formula for more than half a month, and because he had long considered completing a paper, Qiao Yu had prepared the entire derivation process in great detail. The next step was to integrate those derivation processes together using professional language.

It is nothing more than lemmas and theorems, and the proof part can basically be copied and pasted directly.

The main thing is the subsequent proof process of the unified geometric constraint parameter θ, which needs to be written now.

Fortunately, Qiao Yu had a whole weekend to complete the paper.

In fact, there is no need to be so anxious. At Qiao Yu's age, there is no need to rush. It doesn't matter if the thesis is completed a few days earlier or later.

Anyway, he does not need to evaluate his professional title, and he does not have the pressure of 3+3. Even if someone in the world of mathematics does the same research as him and publishes it first, it will not have much impact on him. He can just think about it in a different topic.

After all, there is no pressure to publish papers when you are a student.

But Qiao Yu’s idea is very simple.

He did not read any books or papers this week, so naturally he could not write any reading notes over the weekend, and naturally he could not hand them in to his tutor and senior master's teacher on Monday.

Of course, if he explained it clearly, he believed that both Director Tian and the master on the opposite side would believe him.

But how can a simple explanation be as convincing as directly throwing the paper into the mailbox of your supervisor or senior teacher?

First send the paper to your email, then explain on WeChat: "I'm sorry, tutor, I didn't study as planned last week, so I can't send you my reading experience this week. The reason is that I spent all my time last week perfecting my paper. The paper has been sent to your email."

Qiao Yu didn’t know what other people were thinking, but he had understood one thing since fifth grade: actions are far more convincing than words, and actions that produce results are far more convincing than actions that produce no results.

Trust is either built through fruitful actions or consumed through empty talk or fruitless actions.

Why did he have a stable cash flow starting in the sixth grade? It was because he spent the better part of the fifth grade building up a good reputation and full trust among the primary school students, thus accumulating a fixed customer base of primary school students.

The teacher could not find out the homework he wrote or the signature he signed for others; and he never failed to do so whenever he promised to take an exam for someone to get a high score.

In order to achieve this, he even gave up on his grades. He wrote his homework carelessly and got a C every time. He also relied on guessing for exams, filling the entire paper with answers but still could only get low scores.

He never skipped classes or made trouble. He listened carefully in class and was busy doing homework for others during self-study. This was to give the teacher the impression that he was diligent and obedient, but his grades were not good enough.

The purpose is that even if some students are blind and secretly report it to the teacher, the teacher will not believe that he is capable of helping others with homework and exams...

Now we are just using the same method in school in reverse.

In fact, Qiao Yu has always been like this, meticulous, goal-oriented, and highly proactive. Coupled with his decent IQ, he can naturally get twice the result with half the effort.

So Qiao Yu completed the writing of the paper on Sunday afternoon, and then it was time for a second check and revision of the paper.

After reading the theses of Lao Xue's master's students, Qiao Yu realized how ridiculous some detailed errors in the theses would be in the eyes of reviewers.

Therefore, Qiao Yu also has high requirements for papers. At least, he does not allow obvious inappropriate wording, ambiguity caused by grammatical errors, and logical loopholes. The content of the literature review and the citation format must all be met with unified standards and meticulousness.

Still working until late at night, Qiao Yu finally reviewed the paper twice and made some minor revisions. Only when he was satisfied did he send the paper entitled "Derivation of upper bounds of rational points of algebraic curves based on quasi-complete space, modular form and p-adic geometry" to the two big bosses Tian Yanzhen and Yuan Zhengxin.

Based on the young man's desire to show off, Qiao Yu originally planned to send a copy to Professor Zhang Shuwen of Princeton as well.

But after thinking it over, he gave up. Although he was quite confident, it was his first time to write a paper on such a big topic, and he had to complete it entirely on his own. He was not sure whether his final result would be recognized.

It’s better to let his tutor and grandmaster give him some advice first.

If there are any major problems, he still has a chance to revise it. If Professor Zhang finds any major problems, he will be lectured by outsiders for no reason.

Well, being taught a lesson by one's own mentor can help strengthen the relationship, but being taught by an outsider, when the other person is doing it for one's own good, one can only endure it with frustration.

Not to mention that both my tutor and my grandfather have a good relationship with Professor Zhang. If they approve of this paper, Professor Zhang will definitely see it. It is only a matter of time. There is no need to be anxious.

After submitting the paper, Qiao Yu hurried out, ran to the second floor to take a comfortable shower, returned to the room, casually blew his hair, and then fell asleep.

I didn't feel tired physically this week, but I used up too much mental energy and felt sleepy easily.

……

Huaqing University, Qiuzhai.

Yuan Zhengxin walked into the office at 7:50 as usual, then turned on his computer and logged into WeChat. He planned to print out the book review that Qiao Yu had sent him and take a closer look.

The old gentleman had become accustomed to Qiao Yu writing down his thoughts and insights from his weekly reading and paper-viewing in the form of a document and sending it to him at around 7:30 every Monday morning.

This type of distance education is not a student, but better than a student.

It's easy for people to fall for sweet words, especially the elderly.

Children always like to complain that their elders spend money recklessly, and that all the money they spend on health care and financial management is taken away by outsiders. However, they never reflect on whether they are too perfunctory towards their parents, and whether they take their identities as children as seriously and meticulously as others do.

Excellent salesmen, who ask about the elderly's well-being on a regular basis every day, pay special attention to the emotional needs of the elderly, and can satisfy the elderly in any way as long as it does not cost money, and even go to the elderly's home to massage their backs and feet. They are closer to the elderly than their own sons and daughters, and their words are pleasant to hear. Over time, they can really melt the heart of an iron man...

I was just chatting: "Dad, your illness requires..."

The business will probably be successful, as people's defensive psychology will always disintegrate slowly and invisibly.

Not to mention that Qiao Yu has made some small upgrades here. He doesn't call himself dad, he calls himself grandson...

"Grandpa, I didn't study this week, so I don't have any experience. But your grandson is not lazy. I finished the paper I worked on for a long time this week and sent it to your email. You must help me check carefully to see if there is anything wrong! My goal is to publish in the world's four top mathematics journals!"

This sentence is followed by a serious expression, which fits the character of a grandson very well and seems very naughty.

The old man smiled subconsciously, probably thinking in his heart: His grandson wants to publish in the world's four top mathematics journals. Isn't it enough for you to be proud of publishing in the top section of the journal?

Of course, I am still in a good mood.

After talking with Langjie, the old man downloaded Qiao Yu's paper and read it.

I was very satisfied, especially after I learned from Qiao Yu that the paper was almost written by himself, my evaluation of Qiao Yu naturally increased a bit.

Originally, he was planning to find the corresponding author of the paper, Professor Xue Song from Yujiang University, to inquire about some things, but when he learned that Xue Song had been transferred to Yanbei University, he gave up the idea.

Although he loved Qiao Yu to the core, some barriers were still difficult to let go of.

After all, he is a teacher, and the other party hasn't expressed much yet. Should he be the first to bow his head and admit his mistake to the traitor?

But if it involves Qiao Yu...

Well, anyway, the child recognizes him as his grandmaster and acknowledges that he is from his lineage. That’s enough for the time being.

Soon, the paper was printed out.

The old gentleman had heard Zhang Shuwen talk about Qiao Yu's ideas, so he was not surprised by the title of this paper. But after reading the introduction and table of contents, the old gentleman turned directly to the last two pages and saw the final result. He was stunned and became much more serious.

Wow, this kid actually worked out a general estimation formula?
This also made the old man look more serious and even more excited when reading the paper.

You know what, if this result is correct, at least if there are no obvious problems for the time being, this achievement is really worthy of being published in the four top mathematics journals.

If Qiao Yu's method is correct, it can be applied to numerical applications to effectively assist in judging the convergence of the iterative process, help analyze errors, and optimize the convergence speed of related algorithms. Especially when solving optimization problems, it can help set constraints, ensure that the solution is within the range of rational numbers, and help simplify calculations.

After the old man got serious, he called his assistant into his office and said, "Go and inform them that I will not attend the group meeting this morning. Just give me a copy of the minutes of the meeting."

"Okay, Professor."

……

At Yanbei University, Qiao Yu was doing morning exercises outside his dormitory and study room.

Last week, I was working on some papers and felt like I was welded to a stool every day. It felt like I hadn't had a good activity for a long time, so it was a good time to exercise.

He is only sixteen years old. If he keeps exercising, he can grow taller.

And for Qiao Yu, the new week is like being recharged for seven days, giving him the joy of wasting time.

This is probably the benefit of having a fulfilling life. Even if you are a little lazy occasionally, you can feel very happy.

Unlike those guys who have gotten used to slacking off every day, after all, once people get used to slacking off, the joy it brings becomes less and less, but once something happens and they can no longer slack off, the damage it brings is tons...

When Chen Zhuoyang arrived in front of the building with vigorous steps early in the morning, he saw his junior fellow apprentice doing jumping exercises at the door. He immediately felt like he was seeing youth.

"Qiao Yu, good morning!"

"Brother Chen..., good morning!" Qiao Yu replied, panting.

He felt a little emotional, as he seemed to have not done exercises since he went to college, so it was easy for him to miss the youthful days in high school.

Then I subconsciously remembered that the guy in front of me was only sixteen years old and still in the first year of high school.

"At least I have more good memories of high school life than he does."

Chen Zhuoyang thought to himself with self-mockery, and then suddenly realized that he hadn't been in touch with his high school classmates for a long time. There was a classmate group, but unfortunately few people would talk to each other throughout the year.

In addition, various class reunions have been almost demonized in recent years, and everyone seems to have no intention of getting together. No one is willing to organize such a thankless task. Suddenly, I feel that the memories of those three years in high school are meaningless.

The only thing he missed was probably that his grades were always among the top five in the class, and the envious looks of his classmates every time the teacher announced the results of each exam.

Unfortunately, after coming to Yanbei University, he never felt that kind of feeling of being valued again. Looking back now, Chen Zhuoyang felt that he should have woken up earlier. When he was studying for his master's degree, the classmates around him were already extremely strong, which was completely different from the undergraduate stage.

Really, he felt that he only needed to put in 3% of his effort in undergraduate studies, and his GPA could be consistently ranked in the top % of his grade. When his roommates were still struggling to finish their homework, he could quietly play games beside them, and in between games, he would give his roommates some pointers on their homework, which was part of his daily routine.

People who have never been a top student cannot understand that kind of superiority.

But after being admitted to Yanbei to study for a master's degree, he had to give his all to have a little status among the students taught by his tutor.

As for playing games? You must be kidding!
After he arrived at the research center to study for his doctorate, he was frustrated all the time except for the day he was excited after passing the exam. His goal was to achieve results during his doctoral studies, be recommended to go abroad for exchange, have an academician preside over his doctoral defense, and after graduation, Yanbei University directly offered him a professor or researcher position in order to keep him...

Now, as long as I can graduate, that's enough.

The change in his mental journey has been a lifelong pain for Chen Zhuoyang. All he can say is that the person who was once full of yearning for mathematical research is gone. Many times, Chen Zhuoyang has begun to doubt that he may not be as good as he imagined, because he watched himself gradually fall behind the progress of his classmates.

But since Qiao Yu came, Chen Zhuoyang felt that it was no longer a question of whether he was excellent or not. He began to suspect that he might have a mild mental retardation.

After all, ignorance is bliss. If he hadn't been stupid enough, how could he have had the courage to apply to be Director Tian's doctoral student?

Fortunately, these embarrassing days are about to pass.

If his thesis is passed, he will have two or three months to prepare for the defense and graduate. He can then say goodbye to this place full of talented people and never come back again...

As I walked past Qiao Yu with complicated thoughts in my mind, I suddenly heard a cell phone ringing.

Subconsciously, he pulled out his phone from his pocket, but it wasn't his. Then he heard a young voice behind him.

"Hey, Director Tian, ​​good morning!... Well, I've eaten and I'm working out... Oh, now?... Okay, I'll be right there."

Chen Zhuoyang turned around and asked, "Qiao Yu, the boss asked you to go to the office?"

"Yes." Qiao Yu nodded.

"Oh, do you know what I'm asking you to do?"

"I wrote a paper last week and sent it to my supervisor last night. Professor Tian probably wanted to give me some guidance, right? Or maybe he found some problems with the paper and wanted to explain them to me clearly and send it back for rewrite." Qiao Yu answered cheerfully.

"Oh, you finished your paper? Which one?"

"It's the problem we discussed at the last seminar, about accurately estimating the rational number points on the upper bound of the curve."

"Done it using Schultz's method?"

"Yes, that's right, senior brother, I'm sorry, Director Tian asked me to go there quickly, so I'll leave first. We'll talk later." After saying that, Qiao Yu waved casually at Chen Zhuoyang, then trotted towards the building where Tian Yanzhen's office was located.

Chen Zhuoyang looked seriously at Qiao Yu's quickly departing back, and suddenly felt that it was like his lost youth.

……

"Director Tian, ​​what do you want to talk to me about?" Qiao Yu knocked on the door, but did not go in directly. Instead, he opened the door a crack, stuck his head in, and asked Tian Yanzhen who was sitting at the desk.

He looked as if he saw that the situation was not right and was about to run away.

Tian Yanzhen looked up at Qiao Yu who was making a fuss and was amused: "Come in, close the door!"

"Oh, okay, Director Tian." Qiao Yu then walked into the office obediently and closed the door.

Tian Yanzhen glanced at Qiao Yu, who had nothing in his hands, and said, "Pick a book and sit next to me and read it. I can ask you any questions I have."

After saying that, Tian Yanzhen patted the papers on his desk.

"Okay!" Qiao Yu agreed readily, then came to Tian Yanzhen's bookcase, turned around and asked again: "Director Tian, ​​which book in your bookcase is the most expensive?"

"Huh?" Tian Yanzhen turned his head and looked at Qiao Yu again, not understanding what was going on.

"Anyway, I will definitely not be able to finish reading it today. You will give it to me when I leave. It will be more cost-effective for me to pick a more expensive one. After all, I don’t have many opportunities to come here to read books." Qiao Yu explained seriously.

This sentence made Tian Yanzhen laugh again: "Okay, don't worry about my books. If you want to read them, come and get them anytime. When you graduate, I will give them all to you."

Qiao Yu said excitedly: "Director Tian, ​​I'm serious! I'll come directly to move the books on the day I graduate, and you can't stop me."

Tian Yanzhen said casually: "No problem, just come and move it, it's just a few books, it's not worth much."

Qiao Yu casually pulled out a copy of "Modern Number Theory Classics Guide" from the bookcase and retorted: "Ordinary books are definitely not valuable, but the books that have been placed in your office are different. The value of a book lies first in its author and second in the collector!
Once I get the books, I'll auction them off. These are all books collected by Chinese mathematics masters. Wouldn't their value easily increase by hundreds or even thousands of times?"

Tian Yanzhen was stunned for a moment, then said with a straight face: "Why are you talking so much nonsense? Sit next to me and read a book. Don't talk unless I ask you."

"Oh, okay, Director Tian." Qiao Yu immediately sat down obediently and began to flip through the book.

Flattery is something that should not be done too frequently or too deliberately. It is enough to make the tutor happy once in a while, otherwise the threshold will be raised and it will be troublesome in the future.

As for whether the teacher will buy into this, that is a matter of personal opinion.

But from the perspective of human nature, people all like to hear nice words, especially those from their most proud disciples. As long as he persists and works diligently to pick the most advantageous moments to flatter occasionally, Qiao Yu believes that whether Director Tian has a stern face or not, he must be happy in his heart.

Since the teacher didn't allow him to talk, he didn't argue and just read quietly.

Just a few minutes of silence...

"Are you sure that the three parameters k/p/q are truly independent in this paragraph? Is there enough data to support this?"

"Well, I have derived this and introduced the parameters of several known curves for fitting. Anyway, it seems to be fine for now."

"Manual fitting? You didn't use a supercomputer to do numerical simulation? There are enough data samples in the LMFDB curve database for batch verification. Using a supercomputer to do numerical verification and batch analysis will be more convincing."

"Ah? On a supercomputer?" Qiao Yu was confused. He was just a kid who wanted to publish a paper. Why did he have to rent a supercomputer?
"Oh, you don't have an account for the school's supercomputer yet? Well, you can use my account first. Okay, you continue reading and I'll continue reading."

After saying that, Qiao Yu saw Tian Yanzhen pick up a pen and make a note on the printed paper, then continued to read it.

The rest of the process was unremarkable.

"Are you sure that the relationship between modular form rank k and genus g is generally applicable, especially in the case of high-genus curves or non-trivial modular spaces?"

"It's not just k. This part is actually where multiple tools converge. Professor Zhang said that the proof process can be complicated. The entire logic of this part is based on the multi-level complexity of algebraic geometry. Here I used the theorem of the weak BSD conjecture..."

That’s about it, one question and one answer.

Tian Yanzhen raised the question, and Qiao Yu tried his best to explain his thoughts at the time in detail...

The morning passed by unknowingly, and Qiao Yu only finished reading the paper when he felt his stomach growling.

"One last question, did you complete this paper by yourself?" Tian Yanzhen asked Qiao Yu seriously.

Then, without waiting for Qiao Yu to answer, he took the initiative to explain: “I am not doubting you, but for a paper like this, submitting it after supercomputing verification will be a very serious matter.

If you have discussed with others in various ways, and the other party has indeed given you some effective suggestions or directly participated in the research of the topic, it may be controversial if the paper is published with only your name as the author. In this case, you'd better communicate clearly with the other party, such as whether the other party's name needs to be added. "

There was no need to think about it. Qiao Yu answered directly: "Yes, it is true. I have only discussed this paper with Qiao Xi. If it can be published, I originally planned to add her name as the second author."

"Qiao Xi?" Tian Yanzhen was stunned for a moment, then immediately asked, "Your mother?"

"Yes, my mother, as for the most important part about simplifying the results, I really couldn't figure out how to effectively unify those parameters, so I called my mother. My mother told me that mathematics is about finding possible hidden logical connections between different conditions.

I was inspired by this sentence and finally came up with the final result. Well…my mother currently has no job. She has been staying at home to take care of me, so I added her as the second author. Is it okay to leave the work unit column blank?

I've read other papers and it seems that the authors all have their affiliations listed after them. If there are no affiliations, will there be any problems?"

Tian Yanzhen closed his eyes and thought for a while, then sighed deeply and said, "Not having a work unit is not a problem, and she is not the first author, so it will be fine as long as she is marked as Independent Researcher or directly write Yanbei International Mathematical Research Center.

I mean you haven’t discussed this paper with anyone else in the mathematics community, including... Mr. Yuan, right?”

"Yes, Teacher Tian. I just received the paper last night, so there will definitely be no controversy in this regard. You can rest assured!" Qiao Yu answered with certainty.

As soon as he finished speaking, the phone rang again. Qiao Yu took out his phone and took a look. His expression was a little embarrassed, but he still said, "Hey, it's Master's call."

"Then why are you hesitating? Go ahead."

Director Tian sat there and said it lightly.

　Thanks to joasse for the reward and encouragement!
　　PS: The 26th day of updating words has been completed!
　　
　
(End of this chapter)