Analytic Resolution of Nonsingularities and the Absolute Remote Abelian Conjecture

"Before I recommend some topics to you, let me first talk to you about number theory."

"Number theory studies the properties of integers—all kinds of properties."

"From my perspective, it's about discovering the various coincidences that emerge when integers undergo different transformations."

"Take prime numbers. Don't you think they're like a kind of coincidence among numbers?"

"Numbers that happen to have only two factors, 1 and themselves, appear continuously throughout the natural number sequence. They look completely coincidental, with no discernible pattern."

"Yet in studying prime numbers, mathematicians have tried to derive patterns from something so seemingly patternless, and for that purpose, they proposed a conjecture."

"Do you know what that conjecture is?"

Listening carefully to Liu Bin's explanation, Xiao Yi did not even need to think before answering.

"The Riemann Hypothesis."

"Yes, the Riemann Hypothesis." Liu Bin said, "Its conclusion allows prime numbers, which seem like nothing more than coincidences, to be summarized into a pattern through a function."

"Besides prime numbers, mathematicians have also begun wondering whether the many other coincidences between integers can likewise be distilled into genuine patterns."

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

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

"It also looks like a coincidence, yet mathematicians devoted themselves to proving it."

"Returning to prime numbers, the Twin Prime Conjecture says that among infinitely many primes, pairs of primes differing by 2 will keep appearing. That also sounds like a coincidence, but several years ago... well, in 2013, Mr. Zhang Yitang made a breakthrough. What can now be confirmed is that there are infinitely many prime pairs with a difference of less than 246."

"It looks like we are not far from truly proving the Twin Prime Conjecture."

"Then there is the famous Goldbach Conjecture..."

Liu Bin slowly explained to Xiao Yi all sorts of famous problems in number theory, as well as theories related to number theory.

"...In short, number theory is a field of mathematics that turns coincidences into patterns."

"And precisely because of that, number theory has become highly abstract. Achieving results in it tests one's talent greatly. Researchers need a genius-like instinct, one that can keenly detect patterns among numerical coincidences."

"For undergraduate mathematics, for example, students generally only study Elementary Number Theory. Deeper material is for Graduate Students to learn—and in the end, very few people can persevere with it."

"So before you begin writing a number theory paper, I strongly recommend reading some other papers related to number theory as well. That way, you can absorb some of the thought processes of those top mathematical scholars from their papers."

"I believe you must have gained quite a few insights after reading the paper proving Fermat's Last Theorem, right?"

"Yes," Xiao Yi answered.

In a sense, that paper had brought him more gains than any other paper he had read.

"Mm, that's good." Liu Bin smiled. "In that case, I'll recommend a few topics first. You can choose from among them."

Very quickly, Liu Bin sent over several topics.

"Diophantine approximation of rational numbers with specified parity-check types"

"Frequency problems in the Three Gap Theorem"

"Mean values of arithmetic functions and applications to power sums"

There were quite a few topics. Mathematics professors generally prepared many of them, either for their own future research projects or to provide students with suggestions for paper topics. Thus, Liu Bin directly sent Xiao Yi some of the topics he had prepared for his students.

In the end, Liu Bin also sent Xiao Yi several papers written by leading figures in number theory, so Xiao Yi could read them directly instead of having to search for them himself later.

Having settled all that, Liu Bin said, "All right, that's all I have to say. Take your time choosing from those topics, and read those papers too. They'll all be of great help to you."

"Thank you, Professor!"

Xiao Yi thanked him.

"Haha, there's no need to thank me. Academician Hu said the same thing—if you have any questions, just ask me."

"All right, I need to go teach now, so I'll stop here. And don't put too much pressure on yourself. You don't need to insist on finishing this paper within four months. Honestly, that might be difficult even for mathematics PhD students, so don't push yourself too hard."

In the end, Liu Bin could not help reminding Xiao Yi of this. Otherwise, the pressure on a student would be far too great.

"Mm, I understand."

Xiao Yi expressed his gratitude once more.

Being able to meet a professor like this was, of course, a stroke of luck for him.

After hanging up, he began carefully examining the paper topics before him, searching for something that interested him more.

Very soon, one item made his brows rise.

"Analytic Resolution of Nonsingularities and the Absolute Remote Abelian Conjecture?"

They had just discussed all sorts of "conjectures," which had made him somewhat sensitive to those words.

But the conjectures they had discussed had all been famous ones, while he had never heard of this Absolute Remote Abelian Conjecture.

Of course, Professor Liu had provided an introduction beneath every topic, so Xiao Yi would not be left unable to understand them.

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

Seeing that name, Xiao Yi could not help raising his brows. A good name.

"However, this paper requires extending and developing the conjecture proven by Shinichi Mochizuki, thereby proving..."

Let K and L be finite extensions of Qp, and let X/L and Y/K be two hyperbolic curves satisfying analytic resolution of nonsingularities, such as two hyperbolic Mumford curves. Then the map Isom(X,Y)→Isom(π1alg(X),π1alg(Y))/~ is bijective.

Xiao Yi's brows twitched.

This was the so-called Absolute Remote Abelian Conjecture.

Even for him, it was somewhat difficult to understand?

Moreover, this topic tested not only number theory, but Algebraic Geometry as well.

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

Xiao Yi could not help feeling slightly puzzled.

Could it be that Professor Liu knew I had finished reading all those books, which was why he recommended this topic to me?

However...

It suited him perfectly!

Another chapter will be released shortly.