The top student must be diligent

Section 23

"But why did you choose number theory?"

Xiao Yi said: "I read the paper in which Wiles proved Fermat's Last Theorem, and this paper made me very interested in number theory."

Hearing Xiao Yi's words, Liu Bin immediately became uneasy.

He asked in surprise: "What? Did you read the paper in which Andrew Wiles proved Fermat's Last Theorem? The original paper? Or did you find the proof process on the Internet?"

"The original paper." Xiao Yi replied.

"There are two papers, both published in the third issue of Volume 141 of Annals of Mathematics in 1995."

Xiao Yi sighed: "I see that there are four or five papers in each issue of Annals of Mathematics, but there are only these two papers in that issue. Is this the treatment for a paper that solves a world-class mathematical problem?"

Listening to Xiao Yi's words, Liu Bin "hissed".

Xiao Yi was able to mention "Annals of Mathematics" and to explain that only two papers proving Fermat's Last Theorem by Andrew Wiles were published in that issue, which proved that Xiao Yi had indeed read the original text of those two papers.

And this is more rare than reading those papers that were summarized and simplified in later generations.

This kid... can actually read the original proof of Fermat's Last Theorem?

"Have you finished reading the 40 books I gave you?"

Xiao Yi replied: "I have started reading it for the second time, and I will be able to finish it in a few days."

The speed of reading it for the second time should be faster than the first time, but because he still has to read papers and so on, the speed is much slower.

However, Liu Bin still sighed in his heart.

After reading it for the first time, he is willing to continue reading it for the second time.

This kid really likes mathematics!

Maybe he really has a chance to write a core journal-level paper in his first paper?

Then, Liu Bin said: "Okay, in that case, then I will recommend you some topics in number theory."

Please collect and recommend! !

Chapter 29 Analysis of non-singular points and absolute tele-Abel conjecture

"Before I start recommending topics to you, let me talk to you about number theory."

"Number theory studies the properties of integers, and there are all kinds of properties."

"From my understanding, it is to find all kinds of coincidences that can be found after integers have undergone various transformations."

"Like prime numbers, do you think this thing is like a coincidence between numbers?"

"It just happens that there are only two factors, 1 and itself, which appear in the natural number series continuously. It looks like a coincidence, without any rules."

"However, in the study of prime numbers, mathematicians have tried to summarize the rules of this irregular thing and put forward a conjecture for this."

"Do you know what this conjecture is?"

Listening carefully to Liu Bin's narration, Xiao Yi did not think about this question and said it directly.

"Riemann hypothesis."

"Yes, it is the Riemann hypothesis." Liu Bin said: "The conclusion of the Riemann hypothesis allows prime numbers that seem to be just a coincidence to be summarized into a law on a function."

"In addition to prime numbers, mathematicians have also begun to think about whether there are many other coincidences between integers and whether they can also be summarized into real laws."

"Like the proof of Fermat's Last Theorem you have seen."

"When the integer n\u003e2, the equation x^n+y^n=z^n about x, y, z has no positive integer solution."

"It looks like a coincidence, but mathematicians are committed to proving it."

"Let's go back to prime numbers. The twin prime conjecture states that in an infinite series of prime numbers, there will always be two prime pairs with a difference of 2. This sounds like a coincidence, but a few years ago... well, in 2013, Mr. Zhang Yitang made a breakthrough. Now we can be sure that there are infinite prime pairs with a difference less than 246."

"It seems that we are not far from truly proving the twin prime conjecture."

"For example, the famous Goldbach conjecture..."

Liu Bin slowly told Xiao Yi about various well-known number theory problems and theories related to number theory.

"...In short, number theory is a mathematics that summarizes coincidences into laws."

"Because of this, the abstract level of number theory has become quite high. If you want to make achievements in number theory, it is very challenging to be talented. Researchers need to have a genius sense of smell and be able to sensitively perceive the laws between numerical coincidences."

"Just like for undergraduate mathematics, basically only elementary number theory is studied, and the deeper content is studied by graduate students - even in the end, there are very few people who can persist."

"So, before you start writing a paper on number theory, I also strongly recommend that you read some other papers related to number theory, so that you can also absorb some of their thinking from the papers of other top mathematicians."

"I believe you also have some experience after reading the proof paper of Fermat's Last Theorem?"

"Yes." Xiao Yi replied.

To some extent, this paper brought him the most gains among all the papers he has read.

"Well, that's good." Liu Bin smiled and said: "In that case, I will recommend a few topics to you first, and you can choose from them."

Soon, Liu Bin sent several topic choices.

"Diophantine Approximation of Rational Numbers of Certain Parity Types"

"Frequency Problem of the Three Gap Theorem"

"Application of the Mean Sum and the Power Sum of Arithmetic Functions"

…

There are many topics to choose from. For mathematics professors, they usually prepare a lot of topics, either to prepare for their own future research or to provide suggestions for students to choose thesis topics, so Liu Bin directly mentioned them Some thesis topics prepared for the students were sent to Xiao Yi.

In the end, Liu Bin also sent Xiao Yi some papers written by number theory experts, so that Xiao Yi could read them directly, so that he would not have to look for them himself.

After finishing this, Liu Bin said: "Okay, that's all I have to say. You can choose one of these topics yourself and take a look at those papers. They will be of great help to you."

"Thank you Professor!"

Xiao Yi thanked him.

"Haha, there's nothing to thank you for. This is what Academician Hu said. If you have any questions, just ask me."

"Okay, I have to go to class now, so I won't talk about it for now. Well, you don't have to put too much psychological pressure on me. You don't have to think about finishing this paper within four months. To be honest, this can be difficult even for math PhD students, so you don’t have to push yourself too hard.”

In the end, Liu Bin couldn't help but remind Xiao Yi, otherwise the pressure on a student would be too great.

"Yeah, I know."

Xiao Yi expressed his gratitude again.

It was certainly a blessing for him to meet such a professor.

After hanging up the phone, he began to look carefully at the paper topics in front of him, looking for the content that interested him more.

And soon, something made him raise his eyebrows.

"Analysis of non-singularity points and Abelian conjecture of absolute distance?"

He had talked so much with Liu Bin just now, and he also mentioned various "conjectures", which also made him somewhat sensitive to these two words.

But what I just talked about were some well-known conjectures, and this one was definitely far away from Abel’s conjecture, which he had never heard of.

Of course, Professor Liu gave an introduction under each topic, which prevented Xiao Yi from not understanding these topics.

"This conjecture... actually comes from Grothendieck's Far-Abelian conjecture, but this related conjecture has been proven by this Japanese mathematician named Mochizuki Shinichi."

Seeing this name, Xiao Yi couldn't help but raise his eyebrows, what a good name.

"But what this paper requires is to expand and extend the conjecture proved by Mochizuki Shinichi, so as to prove..."

[Let K, L be the finite extension of Qp, X/L, Y/K be two hyperbolic curves that satisfy non-singular point resolution, such as two hyperbolic Mumford curves, and then map Isom(X, Y)→Isom (π1alg(X), π1alg(Y))/~ is bijective. 】

Xiao Yi's brows moved.

This thing is the so-called absolutely far Abelian conjecture.

How difficult is it for him to understand?

Moreover, this topic not only tests number theory, but also tests algebraic geometry. men and women

"Professor Liu actually recommended this kind of topic to me?"

Xiao Yi couldn't help but feel a little confused.

"Could it be that Professor Liu knew that I had read all those books, so he recommended this topic to me?"

It's just...

Just what he wanted!

…

[Another chapter will be released later]

Chapter 30 College Entrance Examination