The Science Fiction World of Xueba

February 16, 2022.

The sixteenth day of the first lunar month.

The Lantern Festival has just passed.

The campus of Jiang University, which had been silent for a month, became noisy again.

Early in the morning, just as Pang Xuelin arrived at the office, there was a sudden noise outside.

Immediately afterwards, Pang Xuelin's office door was pushed open with a slam.

Perelman, who hadn't seen him for a long time, hurried in.

Zuo Yiqiu also followed closely behind and said, "Professor Pang, I'm sorry, I didn't stop this gentleman..."

Pang Xuelin was slightly taken aback, then laughed, and said, "It's okay, Xiao Zuo, you can go out first."

Then, he turned his gaze to Perlman and said, "Grigory, what can I do for you?"

Perelman looked unkempt, with a beard and curly hair that looked greasy and hadn't been washed in an indefinite amount of time.

He was wearing a brown jacket with black cuffs.

Pang Xuelin hasn't seen Perelman for nearly four or five months. The last time the two met was at the inauguration ceremony of the Pang Xuelin Mathematical Science Research Center of Jiangcheng University. Perelman came over to show his face, and then hurriedly left.

In the past year or so, he has devoted all his energy to the research of Hodge's conjecture.

"Professor Pang, I have proved Hodge's conjecture!"

Perelman waved the manuscript paper in his hand and said excitedly.

"Prove Hodge's conjecture?"

Pang Xuelin was taken aback for a moment, he was very clear about the difficulty of Hodge's guess.

In the interstellar world, when he was trapped by Shu Lao on planet No. 5, he spent more than half a year researching this conjecture, but he never succeeded.

He didn't expect that in the real world, Perelman would solve this conjecture.

"Let me see."

Perelman handed the manuscript to Pang Xuelin.

Pang Xuelin began to flip through the manuscript paper page by page.

Perelman was not in a hurry, and sat down on the small sofa beside him.

After a while, Zuo Yiqiu came in with a steaming cup of coffee and placed it in front of Mr. Perelman.

Afterwards, Zuo Yiqiu quietly closed the office door.

After reading for nearly an hour, Pang Xuelin put down the manuscript, pondered for a moment, and said, "Your method of proof is interesting. Have you shown your manuscript to Xinyi?"

Pang Xuelin just browsed through the manuscript of [UU Reading 00kxs], and probably clarified Perelman's proof ideas. ,

However, the specific proof process still needs careful study.

"not yet."

Perelman shook his head.

Pang Xuelin said: "I'll bring Professor Mochizuki Shinichi here too, let him take a look."

With that said, Pang Xuelin picked up the phone on the table and called Mochizuki Shinichi.

Half an hour later, Mochizuki came to Pang Xuelin's office in a hurry.

Seeing that Perelman was also there, Mochizuki showed surprise on his face: "Say, Grigory, why are you here?"

Immediately afterwards, Mochizuki Shinichi seemed to have thought of something, with an incredulous look in his eyes, and said, "You won't solve Hodge's conjecture, right?"

Perelman has been in retreat during this time, and he knows it.

Today he suddenly came to look for Pang Xuelin, and with Pang Xuelin calling him, Mochizuki guessed Perelman's purpose all of a sudden.

Perelman nodded without speaking.

Pang Xuelin laughed and said, "Xinyi, this is the manuscript of Perlman's proof of Hodge's conjecture. Take a look, too. Is there any problem?"

With that said, Pang Xuelin handed the copy of the manuscript to Mochizuki Shinichi with a touch of warmth.

Just when Mochizuki Shinichi came over,

Pang Xuelin copied the manuscript aside.

"good!"

Mochizuki Shinichi was also polite, took the manuscript, found a chair and sat down opposite Pang Xuelin.

Pang Xuelin also took out a manuscript paper and wrote and drew on it.

The office fell silent.

Both Xuelin Pang and Shinichi Mochizuki were poring over Perelman's manuscript.

Perelman himself sipped his coffee leisurely.

He is a very patient person, even if no one talks to him, he can sit alone for a whole day.

Time passed by, and when it was almost noon, Pang Xuelin called Zuo Yiqiu and asked her to order three takeaways for the three of them.

After dinner, Pang Xuelin and Mochizuki continued to study Perelman's manuscript.

According to Perelman's idea, Pang Xuelin tried to deduce the entire Hodge conjecture proof process from beginning to end.

Before I knew it, it was past three o'clock in the afternoon.

Mochizuki Shinichi finally raised his head and said, "I feel that there is nothing wrong with the overall idea, but further research is needed to infer the details."

Perelman couldn't help but heaved a sigh of relief, with a smile on his face, he turned his gaze to Pang Xuelin and said, "Professor Pang, what do you think?"

Pang Xuelin did not speak, pondered for a moment, and said: "Grigory, come here. On the fifth page of the manuscript, Lemma 3.3.4: ?? is defined on the area Ω in the Riemannian manifold ??4 A smooth function without a critical point. The steepest descent line in the region Ω?? Integral curve. How are you going to solve for level set and steepest descent curvature here?"

Perelman pondered for a moment, picked up a pen, and wrote on the manuscript paper:

[Assume {???1,???2} is the unit orthogonal tangent frame, if ???1 is the unit tangent vector of the curve, then the geodesic curvature of the smooth curve is ??=, where ?? is the curve The arc length parameter of . Since {???1,???2} is the unit orthogonal tangent frame, the geodesic curvature can also be expressed as ??=?\u003c???1, D???2d??\u003e =?div(???2), which is equivalent to saying that the geodesic curvature of a smooth curve is the differential of the unit normal vector of the curve. 】

Pang Xuelin smiled faintly. He couldn't deny Perelman's explanation. He turned to the tenth page and pointed to the above proof: "Then here, in the space form ????, ?? is defined in a strictly convex ring The harmonic function on ??2???1, ?? continues to ??2???1. If ??satisfies ??|????1=1, ??|????2=0, Then, there is |???|(??)\u003e0, ???∈??2???1, and the level set of ?? is strictly convex. How did you give the extremum principle in the last part? "

Perelman continued to explain: [Ω is a bounded connected region in ????, ??∈??2(Ω)????(Ω), consider the operator on Ω??????= ??????(??)????????+????(??)??????+??(??)??……]

"What about here??? is a smooth function on a Riemannian manifold with constant section curvature???????? and???? are the Riemannian curvature tensor and curvature, then ??????=????????+????????????? and ????????=??????? ????2???????????????+????????????+R????????……How about this prove?"

[Take 1≤??,??,??,??,??≤??, 1≤??≤??+1. Take the orthogonal frame field in ???? {???1, ???2, ..., ?????, ?????+1}, where ?????+1 is Outward normal, then {???1,???2,...,???i} is the tangent frame field, and ???=?????+1, the motion equation is...]

...

Shinichi Mochizuki, who was watching from the sidelines, was a little strange. Why did Pang Xuelin always gossip on the Riemannian manifold problem, and asked some relatively simple questions. Some lemmas or definitions were very obvious to deduce.

On the contrary, Perelman didn't show much impatience. He basically explained whatever Pang Xuelin asked.

Time passed by, and before you knew it, more than an hour passed.

Pang Xuelin finally had a clear idea: "Here, from the homology group Hn(M, Z) = 0 of a compact and boundless n-dimensional manifold M, it is deduced that M is not orientable. Then we know from Theorem 4.6.7 that all Even-dimensional projective spaces are non-orientable, and their directional double-covered space is a sphere with the same dimension. Then I would like to ask, the Klein bottle whose directional double-covered is a torus T^2, its space Is curvature a smooth function on the Riemannian manifold?"

When Pang Xuelin said this, not only Perelman was stunned, but Mochizuki Shinichi was also stunned.

This is an extremely subtle logical loophole, from the initial setting to the orientation problem of the four-dimensional Klein bottle, which is equivalent to the basis of the whole process of Hodge's conjecture proof.

If there is a problem with this paragraph, it basically means that the entire proof process has a major flaw.

But that wasn't what shocked Mochizuki.

But Pang Xuelin was able to detect such a subtle logical loophole in such a short period of time.

You must know that Perelman's manuscript has a total of more than 30 pages, and he has omitted many links. If this part of the manuscript is converted into a paper, at least half of the content must be added.

It took Mochizuki Shinichi nearly five hours to read this paper carefully.

In terms of understanding, Mochizuki can only say that he understands Perelman's overall proof idea, and he will spend a few days studying some of the details.

While Pang Xuelin finished reading this paper, he completely understood Perelman's proof ideas in such a short period of time, and even discovered very subtle loopholes in it.

The amazing thinking ability and mathematical intuition displayed here are somewhat beyond Mochizuki Shinichi's imagination.

Under normal circumstances, there is not much difference between top mathematicians like Perelman and Mochizuki in terms of thinking ability alone.

What really reflects the gap between mathematicians is to see whether the other party has creative thinking and can open up a new battlefield in fields that others have never thought of.

And this point requires long-term accumulation and an occasional flash of inspiration.

Mochizuki Shinichi originally thought that even if there was a gap between himself and Pang Xuelin, at least in terms of logical thinking ability, there was no qualitative difference.

But today, Pang Xuelin's performance completely exceeded his imagination.

Where did this monster come from?

Perelman also realized this, but he didn't think so much at this time.

He took the manuscript of the thesis from Pang Xuelin, and deduced it from beginning to end.

The final result proved that Pang Xuelin was correct.

Perelman couldn't hide the look of disappointment on his face. After all, he spent so much effort, but in the end his previous efforts were wasted because of a small loophole. It is really unacceptable.

However, he quickly adjusted his mentality.

In the world of mathematics, after a research result is published, it is normal to be picked on for loopholes.

Just like Andrew Wiles back then, when he proved Fermat's last theorem, he was also picked out by the academic circle.

It's just that it took him another year to fill in this loophole before he proved Fermat's last theorem.

Mochizuki Shinichi is even better at this.

In order to prove the ABC conjecture, I invented a set of Tessie Miller theory of the universe. As a result, no one in the academic circle could understand it, and I have been arguing for more than ten years.

If Pang Xuelin hadn't been born later to prove this conjecture, maybe Shinichi Mochizuki would still be arguing with people in the mathematics world.

"Pang, if I have nothing else to do, I'll go back first. I have to think about it. Is there any way to remedy this loophole."

The three chatted for a while, and Perelman took the initiative to say goodbye and leave.

Seeing Perelman's back disappear behind the door, Mochizuki asked curiously, "Pang, do you think Perelman can prove Hodge's conjecture?"

Pang Xuelin shook his head and said, "I don't know. Let's see if Perelman can fix the loophole. At least in terms of the overall direction of thinking, I don't think there is any problem. By the way, how is your research during this time?" gone?"

Since the ABC conjecture was proved, Mochizuki turned his research direction to the field of continuum potential.

The so-called continuum potential is very simple to express. It refers to how many real numbers are contained in the set of real numbers? In other words, how big is the potential of the set of real numbers?

The determination of the continuum potential is the oldest, most basic and most natural problem in set theory.

For (infinite) sets, the necessary and sufficient condition for two sets to be equipotential is that there is a one-to-one correspondence or bijection between them.

It is well known that natural numbers can be used as a measure of the number of elements contained in finite sets: the necessary and sufficient condition for two finite sets to be equipotential is that they contain the same number of elements.

Therefore, the cardinality of every finite set is uniquely determined by a natural number.

Similarly, the cards of infinite sets are uniquely determined by a base ?α.

The smallest infinite base is ?0, which represents the cardinality of the set composed of all natural numbers.

The first base after ?0 is ?1, the first base after that is ?2, then ?3, etc...

Generally speaking, the base number following the base number ?α is ?α+1: the comparison of the sizes of the two base numbers ?α and ?β is uniquely determined by the length of their subscripts (ordinal numbers α and β).

Every natural number n is a base number smaller than ? 0. For infinite base numbers, ?0\u003c?1\u003c? 2\u003c?3\u003c...

Tor proved in December 1873 that the cardinality of the set (ie continuum) composed of all real numbers is at least -1.

Now the question arises: Which base? Is α the cardinality of the continuum?

Is it ?1? or ?2, ?3, or something else? α?

Tor once conjectured that the cardinality of the continuum is the first uncountable cardinality -1.

This is the tor continuum conjecture, and it is also the first of 23 questions raised by Hilbert in 1900.

Mochizuki shook his head and said with a wry smile: "I just have an idea now, and it may take a long time to really understand this problem."

Then, Mochizuki Shinichi chatted with Pang Xuelin about the recent Ponzi geometry seminar, and then left.