Section 236

His roommates were noncommittal about this.

After all, Xiao Yi had not yet served as a graduate student tutor, and they could not find any information on the relevant website.

So instead of betting on this, they might as well honestly choose better schools.

For example, Lupin, Lupin and Chen Muhua were also roommates, and he has now confirmed that he will go to Princeton University for graduate studies.

But in the end, what they never expected was that Chen Muhua actually won the bet.

Xiao Yi actually planned to accept students.

"Don't mention it, you deserve to die!"

Hearing that this kid was still shouting now, several people started to curse for a while.

Chen Muhua shrugged and said, "This is what I deserve. As a warrior, what I get is in line with the bet I made at the beginning."

"You are not a warrior, you are a gambler."

Lupin rolled his eyes and said.

Chen Muhua laughed, "How can a child cry every day, how can a gambler lose every day, there is nothing you can do, you can't envy such things."

At this moment, he received a message, and after taking a look, his eyes lit up.

"Okay, okay, Xiao Shen asked me to enter the conference room, you just watch me and Xiao Shen chatting and laughing."

He snapped his fingers, then put on his glasses and entered the online conference room.

And Xiao Yi was already in it.

Of course, the camera was also turned on.

Looking at the face on the screen that was as young as himself, Chen Muhua couldn't help but sigh again.

Xiao Shen, there is still no change.

Perhaps the only difference is that Xiao Yi's position in the world has become more important.

If the important theoretical conjectures that Xiao Yi had solved before made him respected by people all over the world.

Then the application-related things he has recently developed, such as the epoch-making battery technology of lithium-sulfur solid-state batteries, have truly made him a person that everyone has to pay attention to.

Respect does not necessarily mean attention.

Recalling the first time I chatted with Xiao Yi in the joint exam group, I now have to work hard to show myself and become his student.

"Your name."

Suddenly, Xiao Yi's voice came from the headset.

Chen Muhua was stunned, then replied: "Wei Yimin?"

Xiao Yi pulled the corner of his mouth, then pretended to bow his head and give a score: "Well, deduct 10 points from the impression score."

Chen Muhua's eyes widened immediately, and he quickly said: "Brother Xiao, it's me, Chen Muhua!"

After hearing this, Xiao Yi clicked his tongue twice: "It's really you, kid. You are a student of Beijing University, how can you think of guaranteeing to come to USTC for graduate school?"

"Of course, it's because of my admiration for you, Xiao Shen, that I can't help but choose USTC. Brother Xiao, you are my..."

"Stop, stop." Xiao Yi stretched out his hand to stop this guy from continuing, then bowed his head and pretended to give a score, saying: "Try to get the favor of the tutor by pleasing him, and deduct 10 points from the impression score."

"Hiss~" Chen Muhua took a breath, and then decisively became honest.

"Brother Xiao..."

"Also, during the interview, you have to refer to your position."

Chen Muhua: "Yes, Professor Xiao."

Xiao Yi smiled. He still had to treat this video conference as an interview as seriously as possible.

After confirming that the guy in front of him was the Chen Muhua, he also opened the guy's information again.

"Okay, since this is an exchange meeting arranged by the school, it can also be regarded as an interview for you, so we won't waste so much time talking and just start the formal part. Please introduce yourself first."

"Okay." Chen Muhua nodded, and then he became serious.

Don't look at the laughter just now, as if he wanted to play the relationship card with Xiao Yi, but in fact he never had such an intention.

He believed that he must be one of the best among all the students who wanted to choose Xiao Yi as a mentor.

And since he had this ability, why did he need to do unorthodox methods?

"Hello, Professor Xiao. I am Chen Muhua. I am a student of the School of Mathematical Sciences of Shangjing University. My average GPA for three years is 3.88. With various bonus points, my total GPA is 4.27."

"In addition, I have independently completed three papers in the past three years, and all of them have been published, including two Chinese core papers and one international second zone SCI."

"At the same time, I have participated in two Qiu Chengtong College Student Mathematics Competitions and won two individual first prizes. In addition, I have participated in the Chinese Mathematical Society and was invited to report once..."

Introducing oneself is naturally the time to introduce various achievements made in the past.

Listening to Chen Muhua's many achievements, Xiao Yi also raised his eyebrows slightly.

It can be seen that Chen Muhua did not waste his talent and time during his three years of undergraduate studies.

Obviously, everyone is working hard in their own world.

As his introduction was completed, Xiao Yi nodded.

"Very good, I can see that you really work hard."

He looked at the titles of the three papers in the information and said, "Well... the research directions of your three papers are all algebraic geometry, and you seem to be very interested in algebraic geometry."

"Yes." Chen Muhua nodded, "Algebraic geometry is one of the most important directions in modern mathematics. At the same time, I am very interested in the Langlands program, so I also want to develop research in number theory."

"Langlands Program."

Xiao Yi nodded slightly, and after thinking for a moment, he asked: "What do you know about number theory now?"

Chen Muhua: "At present, I have studied books on analytic number theory and algebraic number theory. Well... I have basically understood the basic concepts of algebraic number fields and algebraic integers, studied some basic properties of unit groups and class numbers, and read several papers. a"

Xiao Yi nodded slightly.

He thought for a moment and said: "Then I will give you a problem next, you try to solve it."

[Let Q be the field of rational numbers, L be a finite extension field, and Gal(L/Q) be its Galois group. Let π be an automorphic representation on GL2(Q), and ρπ be the representation of Gal(-Q/Q) associated with it. Prove that if π is an automorphic representation corresponding to a modular form, then ρπ is continuous, and its trace function tr(ρπ(Frob_p)) satisfies a_p=tr(ρπ(Frob_p)) with the p-coefficient a_p of the modular form, where Frob_p is the Frobenius element of prime p. 】

After writing the question, Xiao Yi continued: "This question has a deep relationship with the Langlands Program. Since you want to study the Langlands Program, this question may give you some understanding."

"By the way, I'll give you a hint. To solve this problem, you must first pay attention to the relationship between modular forms and automorphic representations, such as the relationship between the Hecke eigenvalues ​​on GL2 and the representation of Frobenius elements."

"Secondly, there is the Langlands correspondence, and the two concepts of Frobenius elements and trace functions are used emphatically. Think carefully about what they do."

"Okay, let's get started."

Listening to Xiao Yi's words, Chen Muhua was already a little dazed at this time.

How could Xiao Shen... come up with such a complicated question?

But considering that this is a question about the Langlands Program, it is normal to be complicated.

It involves algebraic geometry, number theory, and representation theory, the three major components of the Langlands Program.

He looked at it with a slight headache and tried to solve the problem.

And Xiao Yi also began to wait.

Time passed slowly.

10 minutes later.

Chen Muhua still gave up.

He shook his head and said, "Sorry, Professor Xiao, I can only solve this now. I still don't know enough about representation theory."

Xiao Yi smiled, and he was not surprised by this.

"It's okay. Strictly speaking, the difficulty of this question is at the doctoral level. It doesn't matter if you can't solve it, but your previous steps are correct."

"First, let f be a modular form with weight k, and its Fourier expansion is f(z)=∑(∞, n=1)ane^(2πinz). The key point of this step is the relationship between modular forms and automorphic representations. It's good to be able to think of this."

"The following is the correspondence between automorphic representations and Galois representations. This is the key content of representation theory. It requires the accumulation of knowledge. It's normal that you don't understand it."

"Let's use the Langlands correspondence. Each modular form f corresponds to a two-dimensional Galois representation ρπ."

[ρπ:Gal(-Q/Q)→GL2(Ql), where l is a prime number. This Galois representation is continuous and closely related to the modular form f. 】

"Then it's..."

Xiao Yi slowly explained to Chen Muhua the solution process of this problem.

Although some of it was still unclear, Chen Muhua also roughly understood the method of proof.

"...Finally, let's summarize the whole process. The key points are the three tips I just gave, and then the breadth of knowledge."

"For example, we need to use the Hecke eigenvalues ​​of modular forms, the Langlands functor, and finally the work of Deligne."

"In the past, Deligne proved the relationship between the p-coefficients of modular forms and the traces of Galois representations. Specifically, for a new form f with a weight k and its corresponding Galois representation ρπ, we have: ap=tr(ρπ(Frobp))."

"So, if you want to study the Langlands Program, the amount of knowledge is very important, and if you can master so much knowledge and use it without any obstacles, then you can become a real Langlands Program mathematician."

Listening to Xiao Yi's words, Chen Muhua couldn't help but feel excited.

Become a real mathematician?

For those who study mathematics, becoming a mathematician is also what they yearn for most.