God's Crown
Chapter 151 BSD Conjecture (1)
What happened? An Yan a few days ago was delighted and frowned for a while.It's not as exaggerated as it is now. Today is too exaggerated. Seeing An Yan's expression, he always felt that something was about to explode.But he refrained from asking An Yan.
Probably because An Yan was still in the normal category a few days ago, maybe because his stuff was too difficult to understand or this stuff was really too difficult. When he was doing the problem, he broke down a bit.These are all very normal things, so he didn't ask An Yan, but An Yan started off fine today, but his hands began to tremble slightly as he wrote—what's the matter?
Kai and Shinya were extremely curious, glanced at An Yan, and asked in confusion, "An, what's wrong with you?"
"Okay." An Yan breathed a sigh of relief, patted Kai and Shinya on the shoulders and said, "I'll go back first, maybe I won't come to the library for a few days."
"Ah?" Kai and Shinya looked at An Yan's leaving figure, a little puzzled, what did An Yan's words mean just now?If he doesn't come to the library recently, what does he want to do?Kai and Shinya couldn't figure it out.Scratching my head and not knowing what to say, after a while.An Yan had completely disappeared before his eyes. After thinking about it for a while, he still didn't quite understand An Yan's words.
Forget it, don't think about it, the more you think about it, the more you can't figure out why An Yan would say such a thing.
Shaking his head, Kai and Shinya continued to do their own thing.
Go back to the room and turn on the computer.The corners of An Yan's mouth turned up slightly, as if she was very happy about something.After the computer was turned on, An Yan couldn't wait to open the document.
[Hilbert space...
……
Abstract: A new iterative algorithm is proposed and studied for solving the generalized splitting equation problem in the set of infinite-dimensional Hilbert space. It is proved that the sequence generated by the proposed algorithm converges strongly to the generalized splitting equation problem. A Solution and Fixed Points of a Family of Direction Operators...
……
Introduction: H1, H2 are real Hilbert spaces, C, Q are two non-empty closed convex subsets of H1, H2. The splitting feasibility problem is to find a point x
……
H→H is an operator, the fixed point set of T is not empty, for any z∈Fix(T) and x∈H has the following inequality...①】
While An Yan was thinking, he was writing calculations in the document, and he didn't seem to notice the passage of time.He sat in front of the computer and typed his thesis intently.When I was sleepy, I fell asleep.
For several days in a row, Hai and Shinya never saw An Yan's person at all, and they didn't know what An Yan was doing.
After Kai and Shinya finished their work, they decided to go and see An Yan.I don't know what this guy is doing, and he hasn't shown up for several days.Nothing will happen, right?After Kai and Shinya packed their things, they headed to Anyan's dormitory.
Because Professor Hudson is a free-range model, apart from having some friendship with An Yan in the sea and Shinya, no one seems to care where An Yan went and what happened to him.
When they came to An Yan's door, Kai and Shinya knocked on the door.
Then began a long wait. After waiting for an unknown period of time, there was still no sign of anyone opening the door.Kai and Shinya are a little flustered right now, damn it!Nothing really happened to Mr. Yan, right?Reminiscent of the news that some college students suddenly fell ill in the dormitory a few days ago.Kai and Shinya were overwhelmed with anxiety, he was clutching his forehead, walking back and forth outside the door, not knowing what to do now.
Call the police?Don't be funny, if you call the police in the United States, do you want those policemen to shoot and kill Yan Jun?A possible future mathematician and physics master died at the hands of brutal American police?After thinking about it, Kai and Shinya had a shock, but there was nothing he could do now.
God knows what he should do now?Yan Jun has not opened the door, which really makes people feel very panic.
Stretching out his hand and knocking on the door again, Kaihe Shinya cleared his throat and raised his voice, "Yan-kun, are you there?"
'Kacha' the door was pulled open, An Yan stared at Kai and Shinya and said, "Sorry, I fell asleep just now, what's wrong? Shinya-kun, what's the matter?"
"Thank God, I thought something happened to you." Kai and Shinya said with a sigh of relief, "I see that you haven't been able to come to the library for several days. At first I thought you had something to do, and then half a month later I didn’t even come to the library in time. Seeing that it’s almost the end of April, if you don’t show up, I think I might have to force my way in.”
"I'm sorry, Shinya-kun, I've been busy with some things recently. It's just a little bit away, and I'm done with this matter." An Mingyang smiled gently, "So just give me a few more days."
"Alright Mr. Yan." Seeing that An Yan was fine, Kai and Shinya were instantly relieved.He breathed a sigh of relief and said, "I just came to see how you are doing. I'm much more relieved that you're fine."
"Don't worry, I have nothing to do." An Yan naturally didn't have any thoughts about Kai and Shinya, but it was something that moved An Yan to have friends who cared about him.
"Then, Lord Yan, I will take my leave first."
After An Yan gently waved to Kai and Shinya, he closed the door.
Looking at the document on the computer, after thinking for a while, he opened the email.While thinking, he sent a message to Faltings——
[Dear Mr. Faltings, I can’t proceed with the Tate conjecture you mentioned before because there are counterexamples.But when I was studying Abelian varieties, I discovered an important problem-no, or rather, I made a major discovery.
This is a discovery that astonished me so much that I would like to describe it to you in detail if possible.Now I still need more time to study, I think I will be able to send out this Abelian cluster soon. 】
After writing the email, An Yan sent it to Faltings' email.Faltings is very familiar with the issue of Abelian clusters, or in other words, Faltings is indeed an expert in this regard.
Faltins first proved the Tate conjecture, and then derived the Safalevich conjecture about Abelian varieties from the Tate conjecture, which also proved the Safalevich conjecture and the Modal conjecture about curves. ②
Solving several conjectures in one breath, this is a shocking genius mathematician.
After An Yan finished writing the letter to Faltings, she turned on the computer and continued to think.On the issue of Abelian clusters, he did encounter some thorny difficulties.
If Faltings hadn't suggested him to look at Tate's conjecture, maybe he wouldn't have done anything about Abel clusters because of Tate's conjecture.
The so-called Abelian variety is the complete group outline of the geometric whole on the field, which must be projective, smooth and commutative.An algebraic group that is at the same time a complete algebraic variety.
That's why An Yan spent so much time calculating Abelian clusters, so she didn't leave her room.When he was completely selfless and doing things, he never thought about what to do if everyone was in a hurry if he didn't show up for so long.But come to think of it, he doesn't have many friends in America either.Apart from Kai and Shinya, there seemed to be no one else.
After the email was sent, An Yan did not expect that Faltings would reply to his message immediately.After all, he is still a big boss for many years, and after seeing it, it may be possible in a year or two.
Now he is not sure whether he can really make some results on Abelian clusters, which is a very difficult thing.How to describe it?It seems that there is a certain difference between knowing how to make a thing and actually starting to make a thing.
An Yan grabbed her own hair, but she didn't know what to do now.It is really a very difficult choice to continue working on Abelian varieties, or to study your own gauge theory instead.His professor hopes that he will put more thought into high-energy physics. Although there are experiments in high-energy physics at present, it still belongs to the category of theoretical physics in general.
This research is so advanced that it has almost no effect on the current development.And the development of basic disciplines, especially this kind of theoretical stuff, maybe one day in the future, humans will be able to come in handy?
What's more, the ultimate problem of physics now - the connection between grand unified theory and high energy physics is very close.
His professor naturally wanted him to study physics instead of mathematics, especially... spending too much time on mathematics research.Otherwise, his professor will also wonder whether this guy is studying for a Ph.D. in physics or mathematics.
However, Abelian varieties are complete algebraic varieties.The application of algebraic varieties is linked to quantum mechanics, but An Yan scratched his head.Although he told Mr. Faltings that it was still a little bit worse, but it was just a little bit, and he couldn't think of anything.It's just too disturbing.
Alas, I don't know when this thing will be able to be calculated.An Yan rubbed his eyebrows, he felt a little pain now, and he didn't know why, all in all, it was very uncomfortable.
It was only a little bit short, and he was able to make some beautiful things on the Abelian cluster.But just such a little bit made him almost collapse, he couldn't figure it out, couldn't do it.My brain is really hurting... Fortunately, he has created the previous Hilbert space and his Angle space, and has published it on arXiv.
I believe that the group of people in the New York laboratory will be very happy when they see it.After all, he helped them solve a big problem, didn't he?When An Yan thought of this, she snickered.
I don't know what kind of expression the guys in the New York laboratory will have when they discover his paper?
After thinking about it, An Yan felt that his brain was about to explode.Things about Abelian varieties and algebraic varieties had made him feel like he might collapse at any moment.He just had a hard time thinking about the idiots in the New York laboratory, the corners of his mouth slowly turned up, and then he started to do his own business seriously.
It is obviously impossible to complete the Abelian cluster in a few days.An Yan is now facing a huge challenge, and it is not even clear whether he will be able to succeed in his lifetime.
Even Mr. Faltings didn't seem to have made this stuff.
…………
In the New York laboratory, the sunny spring makes people feel refreshed.Sure enough, only in spring can we enjoy such a beautiful morning.Rick turned around and walked into the laboratory.At this time, everyone is still doing various calculations on the Hilbert space.
He smiled slightly and said, "Good morning, everyone."
No one responded to him, sitting in front of the computer.Rick felt that they should be able to finish this project in the spring of next year, and at that time, they would definitely give that Ann a heavy blow.
Just thinking about it makes me feel very happy.
"Oh my god, do you need a cup of coffee?"
Still no one responded to him, well, is this group of people studying stupidly?Not even coffee.However, thinking about it, he has come to the laboratory very early, and I didn't expect that there would be someone earlier than him.It seemed that these guys hadn't slept and had been staying in the laboratory.He really wanted to ask, are these guys still in good health?Is there really nothing wrong?
Humming, Rick made a cup of coffee.It can be seen that he is currently in a very good mood.It is probably because they have achieved certain results in the Hilbert space and Angler space, so he seems to be a little complacent when he hums the tune of the song.
After making coffee and sitting in a chair, Rick started logging on to the arXiv website.He wants to see if that Ann has published any phased results on arXiv recently, if there is any - that would be the best thing, he would like to see, and in the end it was Ann who made it first .It was still a large group of them who did it first, but there is no need to think about this kind of thing, they must be the first to do it.
Are you kidding me? Are there so many of them not as good as An alone?If this is the case, Rick feels that he may really collapse, not just collapse.Maybe there will be a psychological shadow from now on.Just as Rick was easily searching arXiv, An Yan's thesis was uploaded.
After thinking about it, he decided to tell his professor first.By the way, before studying gauge field theory, he still wants to do some things on Abelian clusters, people still need to have a little dream.What if it comes true?
So An Yan first called Professor Hudson.
The person who answered the phone was Richard, Professor Hudson's assistant and doctoral student of Professor Hudson.
"Hello." Richard's voice came from the other end of the phone, "Is that Ann? Is there anything you want to talk to the professor about?"
"I have made a thesis here, and I want to talk to the professor. By the way, there is one more thing I want to talk to the professor face to face."
"You wait." When he said this, Richard put down the phone.It happened that he was in Professor Hudson's office, "Professor, Ann seems to have something to ask you?"
"Let him come over, I happen to be in the office today." Professor Hudson put on his glasses and said with a hand, "He has something urgent, did he tell you?"
"Um, it's about the thesis, and there's another question that needs to be clarified with you face to face."
Professor Hudson nodded, "Let him come over."
The professor was also a little confused about what An Yan wanted to tell him.It's still about the thesis. Could it be that there is something difficult about this thesis that An Yan can't get a clue about? Do you need to ask him?
Richard held the phone and said, "Ann, I've already told the professor, and the professor asked you to come over!"
"Okay, I'll probably be here soon." An Yan put down his phone, took a deep breath, printed out his thesis, sorted it out, and then opened the door to Professor Hudson's office.
When he came to the office, Professor Hudson was doing his own thing.
Standing in front of Professor Hudson, the professor looked up at An Yan, pointed to the chair next to him and said, "Ann, sit down and talk."
Pulling out the chair and sitting down, the professor said peacefully, "I heard Richard said that you have something to ask me, right?"
"Yes, professor." An Yan nodded slightly, "This is my thesis." An Yan put the newly printed thesis on the table and said, "Professor, I went to Princeton University to give an academic report before, and I had an academic report with Germany. Professor Ligne, Professor Witten, and Professor Faltings exchanged ideas, and got some inspiration. After returning, after several months of hard work, the previously planned topic was completed."
"Huh?" Professor Hudson froze for a moment, "Are you sure you've already made it?"
"Sure." An Yan nodded slightly and said, "I had some inspirations before I came back from Princeton University, and now I finally used them in my thesis."
"I'll read your paper later." Professor Hudson looked at the number of pages in the paper. He couldn't finish reading dozens of pages of papers in a few minutes.And there are a lot of calculations in it, and he has to check them one by one. "There is one more thing to ask me?" Professor Hudson put the paper aside, looked at An Yan and said, "Is it still for NP-complete problems?"
"No." An Yan shook her head, then smiled at the professor and said, "This time it's not a matter of NP completeness."
"So?"
"About the question of Abelian clusters, Mr. Faltings asked me to study graph theory before, and I found some very interesting things about Abelian clusters from graph theory. Professor, please allow me to study the gauge field theory. , first do some questions about Abelian varieties."
"Abelian variety?" The professor pondered, "Abelian variety is an algebraic variety, and the application of algebraic variety is very closely related to quantum mechanics. Moreover, Hodge's theory is very useful in the application of general relativity and quantum mechanics. Widespread, but hard to produce anything."
"Ann, are you sure you want to do a project on Abelian clusters?"
"I'm sure that I have some ideas now. Although I haven't been able to figure it out yet, I believe I can figure it out."
"Now that you've confirmed it..." Professor Hudson looked at An Yan and said calmly, "And since you've already made results before, it's hard for me to say anything more."
"All I can say is, Ann, I wish you luck. And, I think if you haven't done a good job after two years. You have to do the gauge theory thing with me. I'll let you do it." Professor Hudson said seriously, "Ann, I allow you to do the project yourself because I believe you have the ability to solve the project you have proposed. But if the project you have proposed is too difficult, Or it won't work out. I think—I'll let you work with me on a subject of my choice. Do you have any opinion on my decision?"
"Not at all." An Yan said thoughtfully, "Professor, I just want to give it a try. If I really can't solve it, I won't force it."
"It's good that you know." The professor nodded slightly, "
If you have nothing else to do, go back and do your own thing first.I also have to read your paper..." Professor Hudson held the paper and smiled at An Yan, "I may need to read your paper for a few days or even half a month. "
"Professor, then I'll go back first." After solving the problem of the thesis, he can finally easily study things about Abelian clusters.
However, it is still very difficult to rely on his current knowledge to completely solve the problems on the Abel cluster.
After An Yan left, Richard came to the office curiously.Seeing that the professor was reading the paper carefully, he asked, "Professor, what did Andu tell you just now?" Richard didn't regard An Yan as his junior at all.
An Yan is a direct doctor, and he is a doctoral student.In fact, the status of the two is equal, and An Yan's research in mathematics and physics may even be beyond his reach.So it was more like he regarded An Yan as a person of the same level.The reason for curiosity is that since An Yan came to Stanford University, he has almost never appeared, so he is particularly curious about what An Yan came to see the professor for.
The professor chuckled, "Richard, how long have you been studying for a Ph.D. with me?"
"Uh, it's been about two years." Richard was taken aback for a moment. He was preparing for a doctoral project recently, but many of them were rejected by the professor.He is not the same as An Yan. An Yan does his own research, and the professor hardly cares about An Yan.But the professor will take care of him - because An Yan's topics are usually more advanced topics, and the topics he chooses, in the eyes of the professor, are too simple.Not worthy of being a PhD subject.
"Really?" The professor frowned and said with a smile, "You have been here for more than two years."
"..." Richard thought that after the professor talked to An Yan, he seemed to have a particularly weird feeling.He couldn't say what it was like, but he thought the professor was weird.
It should be that An Yan said something to the professor, making the professor feel that he is a bit incompetent as a doctoral student, right?It should be... like this?Richard was still thinking about it, and Professor Hudson said, "I have received the paper that Ann gave me." He raised the paper in his hand and said, "The paper on Hilbert space and Angler space, write It has more than 50 pages. It should use functional analysis, harmonic analysis, and calculations in Euler's theorem. You have been a doctoral student with me for more than two years, and you have not been able to make a single project."
"..." Richard was speechless. Although he knew that the professor would definitely say that, he still felt very ashamed when the professor really said it.I have been studying for a Ph.D. with the professor for more than two years, but I have not been able to do anything.An Yan only stayed at Stanford University for more than half a year, and spent most of the time going to the library to read documents and do projects with herself.
It turns out that he made a project so much earlier than he stayed with the professor.
"His subsequent subject should be some calculations on Abelian clusters, and I'm not quite sure what his specific subject is." Professor Hudson frowned slightly when he said this, "But I have a feeling , he is most likely working on the BSD conjecture."
What?Richard's eyes widened in shock, BSD guess?An Yan is studying the BSD conjecture, true or false?Why did he feel so... unreliable? What is the BSD guess?That's a top-level problem on Abelian clusters, there's no way this guy can solve the BSD conjecture.It won't unravel... well, it won't.
"I gave him two years. If he fails to make some achievements on Abelian clusters within two years, he will study gauge field theory with me."
"Professor..." Richard looked at the professor cautiously, "What if he really made the BSD conjecture?"
"What do you think will happen if he really made the BSD conjecture? A well-known conjecture in the mathematics world has a lot to do with quantum mechanics, and has a lot of promising mathematical knowledge. What do you think?" The professor put down the paper and looked at the theory calmly. Chad.
"Probably able to let him graduate directly from Stanford University?"
"That's right, if he can really make the BSD conjecture. Then it is enough to prove that his knowledge and ability are enough to obtain a doctorate degree, and he can be a doctoral supervisor."
"You mean..." Richard said incredulously, "If he really does it, not only will he be able to graduate with a Ph.D., but he might even be a professor at Stanford University?"
"It's not impossible." Professor Hudson said, "I haven't read many papers, but I can see that Ann's idea is very good. And the logic is clear. I think he is a genius. Maybe, I still underestimated him a little bit before."
"……"underestimate?Not at all, Professor Hudson has very strict requirements for his own students.
For example, him, and several other students.It is extremely difficult to get some praise from Professor Hudson, but Professor Hudson does not seem to be stingy at all in praising An Yan.
Probably because An Yan is better than them, or because in Professor Hudson's view, only An Yan can deserve his praise, but in any case, in Professor Hudson's view, An Yan is very superior.On the other hand, the few of them seem to have become negative teaching materials compared to An Yan.
"Richard." The professor looked at Richard and said seriously, "I don't expect you to be able to write a thesis as quickly as Ann. But at least you have to let me see your efforts. In fact, I also don't quite agree with Ann doing too many mathematical problems. Physics and mathematics are related, but as a physicist, doing too many mathematical problems is indeed suspected of playing number games."
"This is a good start by Witten, isn't it?" It seems that Professor Hudson seems to have some opinions on Professor Witten.
But as a student of Professor Hudson, Richard has no position or ability to question this matter.He could only listen to the professor chattering, "I think An should be like Yang, who can achieve very high achievements in theoretical physics. But he seems to have gone astray. He took the road of Witten, yes I don't want to see it."
"It is undeniable that Witten is a very good physicist. But he really likes to play with numbers. He is indeed much better than me in mathematics. But in physics, who is better? It's really not certain."
"Professor, isn't physics now putting forward hypotheses—according to hypothetical experimental theories—and finally finding the corresponding mathematical theory? Even Professor Witten's M-theory is one of the physical theories that can never be verified. But his outstanding performance in physics is indeed related to his achievements in mathematics."
"That's right." Professor Hudson nodded lightly and said, "But Witten loves to play with mathematics too much. Physics is not a vassal of mathematics. We need to find some formulas in mathematics to calculate physics, but not from mathematics. Physics. Do you understand that?"
Richard half-understood, and Professor Hudson suddenly had some headaches.Talking to An Yan is a very good thing, but how to talk to other students, he always has the feeling that they can't understand what he is talking about?Is it his illusion?
Professor Hudson sighed, "Forget it, Richard, you go out first."
"Hey." Richard was a little flustered when he saw the professor's slightly disappointed expression.I don't know which sentence I said wrong just now, which made the professor feel a little disappointed in me.But what has been said, he naturally cannot take it back.
I can only feel annoyed in my heart, and when I talk to the professor in the future, I must think it through before I speak out.
After Richard left the office, Professor Hudson stood up and looked out the window.He was suddenly disappointed with his students, including An Yan.
He always felt that An Yan was getting farther and farther on the road of playing with mathematics, and he could see that An Yan liked physics very much.But he relies too much on mathematics, just as he said with Richard just now, physics is not a vassal of mathematics.Although physics often needs to use mathematics to express this theory when determining a theory.
But it doesn't need to be done entirely by mathematics. Thinking of this, Professor Hudson feels a little headache.
Rubbing the space between his brows, he sat back in his seat.Holding An Yan's paper, he read it again, forget it, let's stop thinking about these things now.Anyway, thinking of the end, there is no result.
…………
Back in his dormitory, An Yan glanced at the computer.Mr. Faltings has not replied to him yet, probably Mr. Faltings is busy recently.
He had to go to the library again now, and he couldn't do anything behind closed doors.He needed a lot of documents and materials. Holding the draft paper and pen, An Yan left the dormitory again and walked towards the library.
When he came to the library, he sat next to Kai and Shinya.Taking the borrowed documents and studying them carefully, Kai and Shinya didn't care so much, but were doing their own things.Now that he already knew that An Yan was busy with his own affairs, he certainly didn't want to disturb An Yan too much.
After reading it for a while, An Yan took the draft paper and continued to calculate.
Professor Hudson is right, the so-called interesting thing by An Yan is indeed the BSD conjecture.When he was doing graph theory, he referred to Abelian varieties and discovered something on finite fields.Combined with Faltings' previous ideas and calculation methods of unraveling the Tate conjecture and promoting the use of Abelian clusters, An Yan came up with some inspiration on Abelian clusters.
Although he didn't know if these inspirations would allow him to solve the BSD conjecture, he had to try.What's more, the weak BSD conjecture has been unraveled.Now the only remaining puzzle is 2 and the derivation.
Of course, this is without a doubt the most difficult part of it.But An Yan still wanted to try to see if he could complete this difficult problem.If he can complete this difficult problem, then maybe he will be the mathematician standing on the podium at the 14 International Congress of Mathematicians.
As one of the Millennium Prize puzzles, the BSD conjecture mainly describes the connection between the arithmetic and analytical properties of Abelian varieties.
An Yan took this question as his subject because of accident, not aiming at the problem of the Millennium Award.
I believe that it is impossible for every doctoral student studying in the Department of Mathematics to take a difficult problem of the level of the Millennium Prize problem as the topic of his doctoral thesis.
Isn't this funny?How could they solve such a problem.This kind of problem can be solved to a certain extent, so why not become a professor instead of being a doctoral student?
An Yan was writing a draft while thinking about things——
[For each f∈R, the mapping f: V→k; α=(a1,...,an)af(α)=f(a1,...,an)
is a polynomial function of V, and if g∈R
……③]
Kai and Shinya next to him seemed to have finished their work. When they looked back at An Yan, they froze for a moment and said, "Yan Jun, what are you doing?"
"Oh, I'm researching some questions about Abelian clusters." An Yan didn't raise his head.
"Abelian cluster?" Kai and Shinya were taken aback for a moment, "Aren't you studying the problem of Hilbert space?"
"Oh, that question, I have already handed over the thesis to the professor. I talked to the professor, and the professor allowed me to do some research on Abelian clusters first." An Yan said with a smile, "What about you? do what?"
"Functional analysis." Rubbing between his brows, Kai and Shinya said, "I really envy you. You are so energetic. After completing the Hilbert space, you still need to study Abelian clusters. However, this question is not biased towards pure mathematics. Is it?"
"Yes, so the time the professor allows me to study is limited, and I don't know if I can do it yet."
Kai and Shinya didn't know what to say, so they had to keep silent.Watching An Yan keep counting, after thinking about it, it seems that he still has a lot of things to do.
He should continue to do his own thing. Halfway through, he said to An Yan, "By the way, Ann, I have to tell you something."
"Well, tell me." An Yan nodded.
"I may be going back to Neon Country in May."
"Good thing." An Yan said, "How long are you going to play this time?"
"I don't know... But after I go back this time, I may plan to get credits more likely when I go to Stanford University."
"Which professor are you planning to study as a graduate student?" An Yan didn't ask other things, as students like them who came to Stanford University from outside the United States would definitely study as a graduate student or even a Ph.D.
"I don't know." Kai and Shinya also imitated An Yan's appearance, rubbed their brows and said, "I haven't figured it out yet, let's talk about it after I think it over."
"Alright." An Yan hummed, he drew a graph on the draft paper, flipped through the literature, and watched the answer to the weak BSD conjecture.
…………
"Shit!" Rick almost threw the coffee cup in his hand on the ground, he said viciously, "How is that possible?"
"How is it possible?" Wen Maoshi looked at Rick in confusion, "What's wrong with you?"
"Impossible, it's absolutely impossible!" Rick's eyes widened, and he pulled Wen Maoshi next to him and said, "Tell me, this is a dream, right? I must be dreaming, right?"
"What's dreaming?" Wen Maoshi still didn't quite understand what Rick meant, so he could only let Rick shake his body.
Seeing Rick's crazy appearance, Wen Maoshi was stunned.What's going on here?
"Rick, stop shaking, stop shaking
Probably because An Yan was still in the normal category a few days ago, maybe because his stuff was too difficult to understand or this stuff was really too difficult. When he was doing the problem, he broke down a bit.These are all very normal things, so he didn't ask An Yan, but An Yan started off fine today, but his hands began to tremble slightly as he wrote—what's the matter?
Kai and Shinya were extremely curious, glanced at An Yan, and asked in confusion, "An, what's wrong with you?"
"Okay." An Yan breathed a sigh of relief, patted Kai and Shinya on the shoulders and said, "I'll go back first, maybe I won't come to the library for a few days."
"Ah?" Kai and Shinya looked at An Yan's leaving figure, a little puzzled, what did An Yan's words mean just now?If he doesn't come to the library recently, what does he want to do?Kai and Shinya couldn't figure it out.Scratching my head and not knowing what to say, after a while.An Yan had completely disappeared before his eyes. After thinking about it for a while, he still didn't quite understand An Yan's words.
Forget it, don't think about it, the more you think about it, the more you can't figure out why An Yan would say such a thing.
Shaking his head, Kai and Shinya continued to do their own thing.
Go back to the room and turn on the computer.The corners of An Yan's mouth turned up slightly, as if she was very happy about something.After the computer was turned on, An Yan couldn't wait to open the document.
[Hilbert space...
……
Abstract: A new iterative algorithm is proposed and studied for solving the generalized splitting equation problem in the set of infinite-dimensional Hilbert space. It is proved that the sequence generated by the proposed algorithm converges strongly to the generalized splitting equation problem. A Solution and Fixed Points of a Family of Direction Operators...
……
Introduction: H1, H2 are real Hilbert spaces, C, Q are two non-empty closed convex subsets of H1, H2. The splitting feasibility problem is to find a point x
……
H→H is an operator, the fixed point set of T is not empty, for any z∈Fix(T) and x∈H has the following inequality...①】
While An Yan was thinking, he was writing calculations in the document, and he didn't seem to notice the passage of time.He sat in front of the computer and typed his thesis intently.When I was sleepy, I fell asleep.
For several days in a row, Hai and Shinya never saw An Yan's person at all, and they didn't know what An Yan was doing.
After Kai and Shinya finished their work, they decided to go and see An Yan.I don't know what this guy is doing, and he hasn't shown up for several days.Nothing will happen, right?After Kai and Shinya packed their things, they headed to Anyan's dormitory.
Because Professor Hudson is a free-range model, apart from having some friendship with An Yan in the sea and Shinya, no one seems to care where An Yan went and what happened to him.
When they came to An Yan's door, Kai and Shinya knocked on the door.
Then began a long wait. After waiting for an unknown period of time, there was still no sign of anyone opening the door.Kai and Shinya are a little flustered right now, damn it!Nothing really happened to Mr. Yan, right?Reminiscent of the news that some college students suddenly fell ill in the dormitory a few days ago.Kai and Shinya were overwhelmed with anxiety, he was clutching his forehead, walking back and forth outside the door, not knowing what to do now.
Call the police?Don't be funny, if you call the police in the United States, do you want those policemen to shoot and kill Yan Jun?A possible future mathematician and physics master died at the hands of brutal American police?After thinking about it, Kai and Shinya had a shock, but there was nothing he could do now.
God knows what he should do now?Yan Jun has not opened the door, which really makes people feel very panic.
Stretching out his hand and knocking on the door again, Kaihe Shinya cleared his throat and raised his voice, "Yan-kun, are you there?"
'Kacha' the door was pulled open, An Yan stared at Kai and Shinya and said, "Sorry, I fell asleep just now, what's wrong? Shinya-kun, what's the matter?"
"Thank God, I thought something happened to you." Kai and Shinya said with a sigh of relief, "I see that you haven't been able to come to the library for several days. At first I thought you had something to do, and then half a month later I didn’t even come to the library in time. Seeing that it’s almost the end of April, if you don’t show up, I think I might have to force my way in.”
"I'm sorry, Shinya-kun, I've been busy with some things recently. It's just a little bit away, and I'm done with this matter." An Mingyang smiled gently, "So just give me a few more days."
"Alright Mr. Yan." Seeing that An Yan was fine, Kai and Shinya were instantly relieved.He breathed a sigh of relief and said, "I just came to see how you are doing. I'm much more relieved that you're fine."
"Don't worry, I have nothing to do." An Yan naturally didn't have any thoughts about Kai and Shinya, but it was something that moved An Yan to have friends who cared about him.
"Then, Lord Yan, I will take my leave first."
After An Yan gently waved to Kai and Shinya, he closed the door.
Looking at the document on the computer, after thinking for a while, he opened the email.While thinking, he sent a message to Faltings——
[Dear Mr. Faltings, I can’t proceed with the Tate conjecture you mentioned before because there are counterexamples.But when I was studying Abelian varieties, I discovered an important problem-no, or rather, I made a major discovery.
This is a discovery that astonished me so much that I would like to describe it to you in detail if possible.Now I still need more time to study, I think I will be able to send out this Abelian cluster soon. 】
After writing the email, An Yan sent it to Faltings' email.Faltings is very familiar with the issue of Abelian clusters, or in other words, Faltings is indeed an expert in this regard.
Faltins first proved the Tate conjecture, and then derived the Safalevich conjecture about Abelian varieties from the Tate conjecture, which also proved the Safalevich conjecture and the Modal conjecture about curves. ②
Solving several conjectures in one breath, this is a shocking genius mathematician.
After An Yan finished writing the letter to Faltings, she turned on the computer and continued to think.On the issue of Abelian clusters, he did encounter some thorny difficulties.
If Faltings hadn't suggested him to look at Tate's conjecture, maybe he wouldn't have done anything about Abel clusters because of Tate's conjecture.
The so-called Abelian variety is the complete group outline of the geometric whole on the field, which must be projective, smooth and commutative.An algebraic group that is at the same time a complete algebraic variety.
That's why An Yan spent so much time calculating Abelian clusters, so she didn't leave her room.When he was completely selfless and doing things, he never thought about what to do if everyone was in a hurry if he didn't show up for so long.But come to think of it, he doesn't have many friends in America either.Apart from Kai and Shinya, there seemed to be no one else.
After the email was sent, An Yan did not expect that Faltings would reply to his message immediately.After all, he is still a big boss for many years, and after seeing it, it may be possible in a year or two.
Now he is not sure whether he can really make some results on Abelian clusters, which is a very difficult thing.How to describe it?It seems that there is a certain difference between knowing how to make a thing and actually starting to make a thing.
An Yan grabbed her own hair, but she didn't know what to do now.It is really a very difficult choice to continue working on Abelian varieties, or to study your own gauge theory instead.His professor hopes that he will put more thought into high-energy physics. Although there are experiments in high-energy physics at present, it still belongs to the category of theoretical physics in general.
This research is so advanced that it has almost no effect on the current development.And the development of basic disciplines, especially this kind of theoretical stuff, maybe one day in the future, humans will be able to come in handy?
What's more, the ultimate problem of physics now - the connection between grand unified theory and high energy physics is very close.
His professor naturally wanted him to study physics instead of mathematics, especially... spending too much time on mathematics research.Otherwise, his professor will also wonder whether this guy is studying for a Ph.D. in physics or mathematics.
However, Abelian varieties are complete algebraic varieties.The application of algebraic varieties is linked to quantum mechanics, but An Yan scratched his head.Although he told Mr. Faltings that it was still a little bit worse, but it was just a little bit, and he couldn't think of anything.It's just too disturbing.
Alas, I don't know when this thing will be able to be calculated.An Yan rubbed his eyebrows, he felt a little pain now, and he didn't know why, all in all, it was very uncomfortable.
It was only a little bit short, and he was able to make some beautiful things on the Abelian cluster.But just such a little bit made him almost collapse, he couldn't figure it out, couldn't do it.My brain is really hurting... Fortunately, he has created the previous Hilbert space and his Angle space, and has published it on arXiv.
I believe that the group of people in the New York laboratory will be very happy when they see it.After all, he helped them solve a big problem, didn't he?When An Yan thought of this, she snickered.
I don't know what kind of expression the guys in the New York laboratory will have when they discover his paper?
After thinking about it, An Yan felt that his brain was about to explode.Things about Abelian varieties and algebraic varieties had made him feel like he might collapse at any moment.He just had a hard time thinking about the idiots in the New York laboratory, the corners of his mouth slowly turned up, and then he started to do his own business seriously.
It is obviously impossible to complete the Abelian cluster in a few days.An Yan is now facing a huge challenge, and it is not even clear whether he will be able to succeed in his lifetime.
Even Mr. Faltings didn't seem to have made this stuff.
…………
In the New York laboratory, the sunny spring makes people feel refreshed.Sure enough, only in spring can we enjoy such a beautiful morning.Rick turned around and walked into the laboratory.At this time, everyone is still doing various calculations on the Hilbert space.
He smiled slightly and said, "Good morning, everyone."
No one responded to him, sitting in front of the computer.Rick felt that they should be able to finish this project in the spring of next year, and at that time, they would definitely give that Ann a heavy blow.
Just thinking about it makes me feel very happy.
"Oh my god, do you need a cup of coffee?"
Still no one responded to him, well, is this group of people studying stupidly?Not even coffee.However, thinking about it, he has come to the laboratory very early, and I didn't expect that there would be someone earlier than him.It seemed that these guys hadn't slept and had been staying in the laboratory.He really wanted to ask, are these guys still in good health?Is there really nothing wrong?
Humming, Rick made a cup of coffee.It can be seen that he is currently in a very good mood.It is probably because they have achieved certain results in the Hilbert space and Angler space, so he seems to be a little complacent when he hums the tune of the song.
After making coffee and sitting in a chair, Rick started logging on to the arXiv website.He wants to see if that Ann has published any phased results on arXiv recently, if there is any - that would be the best thing, he would like to see, and in the end it was Ann who made it first .It was still a large group of them who did it first, but there is no need to think about this kind of thing, they must be the first to do it.
Are you kidding me? Are there so many of them not as good as An alone?If this is the case, Rick feels that he may really collapse, not just collapse.Maybe there will be a psychological shadow from now on.Just as Rick was easily searching arXiv, An Yan's thesis was uploaded.
After thinking about it, he decided to tell his professor first.By the way, before studying gauge field theory, he still wants to do some things on Abelian clusters, people still need to have a little dream.What if it comes true?
So An Yan first called Professor Hudson.
The person who answered the phone was Richard, Professor Hudson's assistant and doctoral student of Professor Hudson.
"Hello." Richard's voice came from the other end of the phone, "Is that Ann? Is there anything you want to talk to the professor about?"
"I have made a thesis here, and I want to talk to the professor. By the way, there is one more thing I want to talk to the professor face to face."
"You wait." When he said this, Richard put down the phone.It happened that he was in Professor Hudson's office, "Professor, Ann seems to have something to ask you?"
"Let him come over, I happen to be in the office today." Professor Hudson put on his glasses and said with a hand, "He has something urgent, did he tell you?"
"Um, it's about the thesis, and there's another question that needs to be clarified with you face to face."
Professor Hudson nodded, "Let him come over."
The professor was also a little confused about what An Yan wanted to tell him.It's still about the thesis. Could it be that there is something difficult about this thesis that An Yan can't get a clue about? Do you need to ask him?
Richard held the phone and said, "Ann, I've already told the professor, and the professor asked you to come over!"
"Okay, I'll probably be here soon." An Yan put down his phone, took a deep breath, printed out his thesis, sorted it out, and then opened the door to Professor Hudson's office.
When he came to the office, Professor Hudson was doing his own thing.
Standing in front of Professor Hudson, the professor looked up at An Yan, pointed to the chair next to him and said, "Ann, sit down and talk."
Pulling out the chair and sitting down, the professor said peacefully, "I heard Richard said that you have something to ask me, right?"
"Yes, professor." An Yan nodded slightly, "This is my thesis." An Yan put the newly printed thesis on the table and said, "Professor, I went to Princeton University to give an academic report before, and I had an academic report with Germany. Professor Ligne, Professor Witten, and Professor Faltings exchanged ideas, and got some inspiration. After returning, after several months of hard work, the previously planned topic was completed."
"Huh?" Professor Hudson froze for a moment, "Are you sure you've already made it?"
"Sure." An Yan nodded slightly and said, "I had some inspirations before I came back from Princeton University, and now I finally used them in my thesis."
"I'll read your paper later." Professor Hudson looked at the number of pages in the paper. He couldn't finish reading dozens of pages of papers in a few minutes.And there are a lot of calculations in it, and he has to check them one by one. "There is one more thing to ask me?" Professor Hudson put the paper aside, looked at An Yan and said, "Is it still for NP-complete problems?"
"No." An Yan shook her head, then smiled at the professor and said, "This time it's not a matter of NP completeness."
"So?"
"About the question of Abelian clusters, Mr. Faltings asked me to study graph theory before, and I found some very interesting things about Abelian clusters from graph theory. Professor, please allow me to study the gauge field theory. , first do some questions about Abelian varieties."
"Abelian variety?" The professor pondered, "Abelian variety is an algebraic variety, and the application of algebraic variety is very closely related to quantum mechanics. Moreover, Hodge's theory is very useful in the application of general relativity and quantum mechanics. Widespread, but hard to produce anything."
"Ann, are you sure you want to do a project on Abelian clusters?"
"I'm sure that I have some ideas now. Although I haven't been able to figure it out yet, I believe I can figure it out."
"Now that you've confirmed it..." Professor Hudson looked at An Yan and said calmly, "And since you've already made results before, it's hard for me to say anything more."
"All I can say is, Ann, I wish you luck. And, I think if you haven't done a good job after two years. You have to do the gauge theory thing with me. I'll let you do it." Professor Hudson said seriously, "Ann, I allow you to do the project yourself because I believe you have the ability to solve the project you have proposed. But if the project you have proposed is too difficult, Or it won't work out. I think—I'll let you work with me on a subject of my choice. Do you have any opinion on my decision?"
"Not at all." An Yan said thoughtfully, "Professor, I just want to give it a try. If I really can't solve it, I won't force it."
"It's good that you know." The professor nodded slightly, "
If you have nothing else to do, go back and do your own thing first.I also have to read your paper..." Professor Hudson held the paper and smiled at An Yan, "I may need to read your paper for a few days or even half a month. "
"Professor, then I'll go back first." After solving the problem of the thesis, he can finally easily study things about Abelian clusters.
However, it is still very difficult to rely on his current knowledge to completely solve the problems on the Abel cluster.
After An Yan left, Richard came to the office curiously.Seeing that the professor was reading the paper carefully, he asked, "Professor, what did Andu tell you just now?" Richard didn't regard An Yan as his junior at all.
An Yan is a direct doctor, and he is a doctoral student.In fact, the status of the two is equal, and An Yan's research in mathematics and physics may even be beyond his reach.So it was more like he regarded An Yan as a person of the same level.The reason for curiosity is that since An Yan came to Stanford University, he has almost never appeared, so he is particularly curious about what An Yan came to see the professor for.
The professor chuckled, "Richard, how long have you been studying for a Ph.D. with me?"
"Uh, it's been about two years." Richard was taken aback for a moment. He was preparing for a doctoral project recently, but many of them were rejected by the professor.He is not the same as An Yan. An Yan does his own research, and the professor hardly cares about An Yan.But the professor will take care of him - because An Yan's topics are usually more advanced topics, and the topics he chooses, in the eyes of the professor, are too simple.Not worthy of being a PhD subject.
"Really?" The professor frowned and said with a smile, "You have been here for more than two years."
"..." Richard thought that after the professor talked to An Yan, he seemed to have a particularly weird feeling.He couldn't say what it was like, but he thought the professor was weird.
It should be that An Yan said something to the professor, making the professor feel that he is a bit incompetent as a doctoral student, right?It should be... like this?Richard was still thinking about it, and Professor Hudson said, "I have received the paper that Ann gave me." He raised the paper in his hand and said, "The paper on Hilbert space and Angler space, write It has more than 50 pages. It should use functional analysis, harmonic analysis, and calculations in Euler's theorem. You have been a doctoral student with me for more than two years, and you have not been able to make a single project."
"..." Richard was speechless. Although he knew that the professor would definitely say that, he still felt very ashamed when the professor really said it.I have been studying for a Ph.D. with the professor for more than two years, but I have not been able to do anything.An Yan only stayed at Stanford University for more than half a year, and spent most of the time going to the library to read documents and do projects with herself.
It turns out that he made a project so much earlier than he stayed with the professor.
"His subsequent subject should be some calculations on Abelian clusters, and I'm not quite sure what his specific subject is." Professor Hudson frowned slightly when he said this, "But I have a feeling , he is most likely working on the BSD conjecture."
What?Richard's eyes widened in shock, BSD guess?An Yan is studying the BSD conjecture, true or false?Why did he feel so... unreliable? What is the BSD guess?That's a top-level problem on Abelian clusters, there's no way this guy can solve the BSD conjecture.It won't unravel... well, it won't.
"I gave him two years. If he fails to make some achievements on Abelian clusters within two years, he will study gauge field theory with me."
"Professor..." Richard looked at the professor cautiously, "What if he really made the BSD conjecture?"
"What do you think will happen if he really made the BSD conjecture? A well-known conjecture in the mathematics world has a lot to do with quantum mechanics, and has a lot of promising mathematical knowledge. What do you think?" The professor put down the paper and looked at the theory calmly. Chad.
"Probably able to let him graduate directly from Stanford University?"
"That's right, if he can really make the BSD conjecture. Then it is enough to prove that his knowledge and ability are enough to obtain a doctorate degree, and he can be a doctoral supervisor."
"You mean..." Richard said incredulously, "If he really does it, not only will he be able to graduate with a Ph.D., but he might even be a professor at Stanford University?"
"It's not impossible." Professor Hudson said, "I haven't read many papers, but I can see that Ann's idea is very good. And the logic is clear. I think he is a genius. Maybe, I still underestimated him a little bit before."
"……"underestimate?Not at all, Professor Hudson has very strict requirements for his own students.
For example, him, and several other students.It is extremely difficult to get some praise from Professor Hudson, but Professor Hudson does not seem to be stingy at all in praising An Yan.
Probably because An Yan is better than them, or because in Professor Hudson's view, only An Yan can deserve his praise, but in any case, in Professor Hudson's view, An Yan is very superior.On the other hand, the few of them seem to have become negative teaching materials compared to An Yan.
"Richard." The professor looked at Richard and said seriously, "I don't expect you to be able to write a thesis as quickly as Ann. But at least you have to let me see your efforts. In fact, I also don't quite agree with Ann doing too many mathematical problems. Physics and mathematics are related, but as a physicist, doing too many mathematical problems is indeed suspected of playing number games."
"This is a good start by Witten, isn't it?" It seems that Professor Hudson seems to have some opinions on Professor Witten.
But as a student of Professor Hudson, Richard has no position or ability to question this matter.He could only listen to the professor chattering, "I think An should be like Yang, who can achieve very high achievements in theoretical physics. But he seems to have gone astray. He took the road of Witten, yes I don't want to see it."
"It is undeniable that Witten is a very good physicist. But he really likes to play with numbers. He is indeed much better than me in mathematics. But in physics, who is better? It's really not certain."
"Professor, isn't physics now putting forward hypotheses—according to hypothetical experimental theories—and finally finding the corresponding mathematical theory? Even Professor Witten's M-theory is one of the physical theories that can never be verified. But his outstanding performance in physics is indeed related to his achievements in mathematics."
"That's right." Professor Hudson nodded lightly and said, "But Witten loves to play with mathematics too much. Physics is not a vassal of mathematics. We need to find some formulas in mathematics to calculate physics, but not from mathematics. Physics. Do you understand that?"
Richard half-understood, and Professor Hudson suddenly had some headaches.Talking to An Yan is a very good thing, but how to talk to other students, he always has the feeling that they can't understand what he is talking about?Is it his illusion?
Professor Hudson sighed, "Forget it, Richard, you go out first."
"Hey." Richard was a little flustered when he saw the professor's slightly disappointed expression.I don't know which sentence I said wrong just now, which made the professor feel a little disappointed in me.But what has been said, he naturally cannot take it back.
I can only feel annoyed in my heart, and when I talk to the professor in the future, I must think it through before I speak out.
After Richard left the office, Professor Hudson stood up and looked out the window.He was suddenly disappointed with his students, including An Yan.
He always felt that An Yan was getting farther and farther on the road of playing with mathematics, and he could see that An Yan liked physics very much.But he relies too much on mathematics, just as he said with Richard just now, physics is not a vassal of mathematics.Although physics often needs to use mathematics to express this theory when determining a theory.
But it doesn't need to be done entirely by mathematics. Thinking of this, Professor Hudson feels a little headache.
Rubbing the space between his brows, he sat back in his seat.Holding An Yan's paper, he read it again, forget it, let's stop thinking about these things now.Anyway, thinking of the end, there is no result.
…………
Back in his dormitory, An Yan glanced at the computer.Mr. Faltings has not replied to him yet, probably Mr. Faltings is busy recently.
He had to go to the library again now, and he couldn't do anything behind closed doors.He needed a lot of documents and materials. Holding the draft paper and pen, An Yan left the dormitory again and walked towards the library.
When he came to the library, he sat next to Kai and Shinya.Taking the borrowed documents and studying them carefully, Kai and Shinya didn't care so much, but were doing their own things.Now that he already knew that An Yan was busy with his own affairs, he certainly didn't want to disturb An Yan too much.
After reading it for a while, An Yan took the draft paper and continued to calculate.
Professor Hudson is right, the so-called interesting thing by An Yan is indeed the BSD conjecture.When he was doing graph theory, he referred to Abelian varieties and discovered something on finite fields.Combined with Faltings' previous ideas and calculation methods of unraveling the Tate conjecture and promoting the use of Abelian clusters, An Yan came up with some inspiration on Abelian clusters.
Although he didn't know if these inspirations would allow him to solve the BSD conjecture, he had to try.What's more, the weak BSD conjecture has been unraveled.Now the only remaining puzzle is 2 and the derivation.
Of course, this is without a doubt the most difficult part of it.But An Yan still wanted to try to see if he could complete this difficult problem.If he can complete this difficult problem, then maybe he will be the mathematician standing on the podium at the 14 International Congress of Mathematicians.
As one of the Millennium Prize puzzles, the BSD conjecture mainly describes the connection between the arithmetic and analytical properties of Abelian varieties.
An Yan took this question as his subject because of accident, not aiming at the problem of the Millennium Award.
I believe that it is impossible for every doctoral student studying in the Department of Mathematics to take a difficult problem of the level of the Millennium Prize problem as the topic of his doctoral thesis.
Isn't this funny?How could they solve such a problem.This kind of problem can be solved to a certain extent, so why not become a professor instead of being a doctoral student?
An Yan was writing a draft while thinking about things——
[For each f∈R, the mapping f: V→k; α=(a1,...,an)af(α)=f(a1,...,an)
is a polynomial function of V, and if g∈R
……③]
Kai and Shinya next to him seemed to have finished their work. When they looked back at An Yan, they froze for a moment and said, "Yan Jun, what are you doing?"
"Oh, I'm researching some questions about Abelian clusters." An Yan didn't raise his head.
"Abelian cluster?" Kai and Shinya were taken aback for a moment, "Aren't you studying the problem of Hilbert space?"
"Oh, that question, I have already handed over the thesis to the professor. I talked to the professor, and the professor allowed me to do some research on Abelian clusters first." An Yan said with a smile, "What about you? do what?"
"Functional analysis." Rubbing between his brows, Kai and Shinya said, "I really envy you. You are so energetic. After completing the Hilbert space, you still need to study Abelian clusters. However, this question is not biased towards pure mathematics. Is it?"
"Yes, so the time the professor allows me to study is limited, and I don't know if I can do it yet."
Kai and Shinya didn't know what to say, so they had to keep silent.Watching An Yan keep counting, after thinking about it, it seems that he still has a lot of things to do.
He should continue to do his own thing. Halfway through, he said to An Yan, "By the way, Ann, I have to tell you something."
"Well, tell me." An Yan nodded.
"I may be going back to Neon Country in May."
"Good thing." An Yan said, "How long are you going to play this time?"
"I don't know... But after I go back this time, I may plan to get credits more likely when I go to Stanford University."
"Which professor are you planning to study as a graduate student?" An Yan didn't ask other things, as students like them who came to Stanford University from outside the United States would definitely study as a graduate student or even a Ph.D.
"I don't know." Kai and Shinya also imitated An Yan's appearance, rubbed their brows and said, "I haven't figured it out yet, let's talk about it after I think it over."
"Alright." An Yan hummed, he drew a graph on the draft paper, flipped through the literature, and watched the answer to the weak BSD conjecture.
…………
"Shit!" Rick almost threw the coffee cup in his hand on the ground, he said viciously, "How is that possible?"
"How is it possible?" Wen Maoshi looked at Rick in confusion, "What's wrong with you?"
"Impossible, it's absolutely impossible!" Rick's eyes widened, and he pulled Wen Maoshi next to him and said, "Tell me, this is a dream, right? I must be dreaming, right?"
"What's dreaming?" Wen Maoshi still didn't quite understand what Rick meant, so he could only let Rick shake his body.
Seeing Rick's crazy appearance, Wen Maoshi was stunned.What's going on here?
"Rick, stop shaking, stop shaking
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