Analytic and Absolutely Remote Abelian Conjecture of Non-singular Points

"Before I recommend some topics for your paper, let me talk to you about Number Theory."

"Number Theory is the study of the properties of integers, all sorts of properties."

"As I see it, it's about discovering the various coincidences that emerge when integers undergo different transformations."

"Take prime numbers, for instance. Don't you think they're like coincidences among numbers?"

"Numbers that happen to have only two factors, 1 and themselves, appear continuously in the natural number sequence, looking completely like coincidences, with no pattern."

"However, in the study of prime numbers, mathematicians have tried to find patterns in this seemingly patternless phenomenon and have proposed a conjecture for it."

"Do you know what this conjecture is?"

Listening carefully to Liu Bin's explanation, Xiao Yi answered the question without hesitation.

"The Riemann Hypothesis."

"Yes, the Riemann Hypothesis," Liu Bin said. "The conclusion of the Riemann Hypothesis allows the seemingly coincidental nature of prime numbers to be summarized into a pattern within a function."

"Besides prime numbers, mathematicians also began to consider whether many other coincidences among integers could also be summarized into true patterns."

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

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

"This also seems like a coincidence, but mathematicians are dedicated to proving it."

"Returning to prime numbers, the Twin Prime Conjecture states that in an infinite sequence of prime numbers, there will always be pairs of primes with a difference of 2. This also sounds like a coincidence, but a few years ago... well, in 2013, Mr. Zhang Yitang made a breakthrough, and it's now confirmed that there are infinitely many prime pairs with a difference less than 246."

"It seems we're not far from proving the Twin Prime Conjecture."

"And then there's the famous Goldbach Conjecture..."

Liu Bin slowly explained various well-known problems in Number Theory and related theories to Xiao Yi.

"...In summary, Number Theory is a field of mathematics that summarizes coincidences into patterns."

"Because of this, Number Theory becomes highly abstract. Achieving results in Number Theory requires exceptional talent and a researcher with a genius-like intuition, capable of keenly sensing patterns within numerical coincidences."

"For example, in undergraduate mathematics, students typically only study Elementary Number Theory. More advanced topics are usually studied by graduate students—and in the end, very few people can persevere."

"So, before you start writing a Number Theory paper, I highly recommend you read some other papers on Number Theory. This way, you can absorb some of the thinking from those top mathematicians' papers."

"I believe you gained quite a bit from reading the paper on the proof of Fermat's Last Theorem?"

"Yes," Xiao Yi replied.

In a way, that paper provided him with more insights than any other he had read.

"Good, that's excellent," Liu Bin said with a smile. "Since that's the case, I'll recommend a few topics for you to choose from."

Soon, Liu Bin sent over several topic suggestions.

"Diophantine Approximation of Rational Numbers with Specific Parity Types"

"Frequency Problems of the Three-Gap Theorem"

"The Average Value of Arithmetic Functions and Their Application to Power Sums"

There were quite a few topics. Mathematics professors usually prepare many suggestions, either for their own future research or to offer to students for their thesis topics. Therefore, Liu Bin directly sent Xiao Yi some of the paper topics he had prepared for his students.

Finally, Liu Bin also sent Xiao Yi some papers written by prominent figures in Number Theory, so Xiao Yi wouldn't have to search for them himself.

With these matters settled, Liu Bin said, "Alright, that's all I have to say. Please choose from these topics and read those papers; they will be very helpful to you."

"Thank you, Professor!"

Xiao Yi expressed his gratitude.

"Haha, no need to thank me. Academician Hu also said that you can ask me any questions you have."

"Okay, I have to go to class now, so I'll wrap this up. And don't put too much pressure on yourself to finish this paper within four months. Honestly, that would be difficult even for some math PhD students, so don't push yourself too hard."

Liu Bin couldn't help but remind Xiao Yi one last time, not wanting the student to feel overwhelmed.

"I understand."

Xiao Yi expressed his gratitude again.

Meeting such a professor was indeed a stroke of luck for him.

After hanging up, he began to carefully examine the paper topics in front of him, searching for something that piqued his interest.

Soon, one item made him raise an eyebrow.

"Analytic and Absolutely Remote Abelian Conjecture of Non-singular Points?"

Having discussed so many "conjectures" with Liu Bin earlier, he had developed a certain sensitivity to these words.

However, the conjectures discussed previously were all well-known, and he hadn't heard of this "Remote Abelian Conjecture" before.

Of course, Professor Liu had provided a brief introduction under each topic, which prevented Xiao Yi from being completely lost.

"This conjecture... it actually originates from Grothendieck's Remote Abelian Conjecture, but a related conjecture has already been proven by this Japanese mathematician named Shinichi Mochizuki."

Upon seeing the name, Xiao Yi couldn't help but raise an eyebrow. What a name.

"However, this paper requires an extension and expansion based on the conjecture proven by Shinichi Mochizuki, in order to prove..."

[Let K and L be finite extensions of Qp, and let X/L and Y/K be two hyperbolic curves satisfying non-singular resolution, such as two hyperbolic Mumford curves. Then the map Isom(X, Y) → Isom(π1alg(X), π1alg(Y))/~ is a bijection.]

Xiao Yi's brow twitched.

This was the so-called Analytic and Absolutely Remote Abelian Conjecture of Non-singular Points.

For him, understanding it was surprisingly difficult.

Moreover, this topic tested not only 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 puzzled.

"Could it be that Professor Liu knew I had finished reading all those books, and that's why he recommended this topic?"

However...

It suited him perfectly!

[Another chapter will be released later]