The top student must be diligent.

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 is the study of the properties of integers, and there are all kinds of properties."

"From my understanding, it is to discover the various coincidences that can be found after integers have undergone various transformations."

"Like prime numbers, do you think this is just a coincidence of numbers?"

"It just so happens that the numbers with only two factors, 1 and itself, appear continuously in the natural number series. It seems like a complete coincidence, with no pattern at all."

"However, in their study of prime numbers, mathematicians tried to summarize this irregular thing into a pattern and proposed a conjecture for this purpose."

"Do you know what this conjecture is?"

Listening carefully to Liu Bin's story, Xiao Yi answered the question without thinking.

"Riemann Hypothesis."

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

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

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

"For integer n > 2, the equation x^n+y^n=z^n related to x, y, and z has no positive integer solutions."

“It also 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 pairs of prime numbers 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 pairs of prime numbers with a difference less than 246."

“It looks like we are not far from actually proving the twin prime conjecture.”

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

Liu Bin slowly told Xiao Yi various well-known problems in number theory, as well as theories related to number theory.

"…In short, number theory is a kind of mathematics that summarizes coincidences into regularities."

"Because of this, the level of abstraction in number theory has become quite high. To achieve results in number theory, one must have a lot of talent. It requires researchers to have a genius sense of smell and be able to sensitively perceive the patterns between numerical coincidences."

"For example, for undergraduate mathematics, most people only learn elementary number theory, and more in-depth content is learned by graduate students. Even at the end, few people can persist."

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

"I believe you also had some insights after reading the proof of Fermat's Last Theorem?"

"Yes." Xiao Yi replied.

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

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

Soon, Liu Bin sent several topics.

Diophantine approximation of rational numbers of certain parity types

Frequency Problem of the Three-Gap Theorem

The Mean of Arithmetic Functions and Application to Sums of Powers

……

There are many topics. Mathematics professors usually prepare a lot of topics, either for their own future research projects or to provide suggestions for students' paper topics. So Liu Bin directly sent some of the paper topics he prepared for his students 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 and no longer have to look for them himself.

After finishing all this, Liu Bin said, "Okay, that's all I have to say. You can choose from these topics and read the papers. They will be of great help to you."

"Thank you Professor!"

Xiao Yi thanked him.

"Haha, there's nothing to thank me 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, don't put too much pressure on yourself. Don't think about finishing this paper within four months. To be honest, this may be difficult for those mathematics doctoral students. You don't have to push yourself too hard."

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

"Well, I know."

Xiao Yi expressed his gratitude again.

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

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

And soon, something made him raise his brows.

"Analysis of non-singular points and the absolute tele-Abel conjecture?"

I just talked so much with Liu Bin, and mentioned various "conjectures", which made him sensitive to these two words.

However, what we just talked about were some well-known conjectures, and he had never heard of this Absolute Abel conjecture.

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 Abel conjecture, but this related conjecture has been proved by a Japanese mathematician named Shinichi Mochizuki.”

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

"But this paper requires us to expand upon the conjecture proved by Mochizuki Shinichi and prove..."

[Let K, L be finite extensions of Qp, X/L, Y/K be two hyperbolic curves satisfying non-singular resolution, e.g. two hyperbolic Mumford curves, then the mapping Isom(X, Y) → Isom(π1alg(X), π1alg(Y))/~ is a bijection. ]

Xiao Yi's brows twitched.

This thing is the so-called absolute far Abel conjecture.

For him, it is actually quite difficult to understand?
Moreover, this topic not only tests number theory, but also algebraic geometry.

"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 finished reading all those books, so he recommended this topic to me?"

It's just...

Just what he wanted!

……

[Another chapter will be released later]

(End of this chapter)